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{SECT 0 {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 
19 "Matem\341tica Discreta" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
258 "" 0 "" {TEXT 258 34 "Ingenier\355a T\351cnica en Inform\341tica \+
" }}{PARA 259 "" 0 "" {TEXT 259 24 "de Sistemas y de Gesti\363n" }
{TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 260 5 "ESCET" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 27 "Ex\341menes de Maple
 resueltos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Examen de Maple. Tipo A
. Febrero 2002." }}{PARA 0 "" 0 "" {TEXT -1 129 "1.- En el conjunto de
 los primeros mil n\372meros naturales al cuadrado, determinar cu\341n
tos de ellos son congruentes con 1 m\363dulo 7." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "restart; with(numtheory):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 70 "S:=0: for i from 1 to 1000 do\nif i^(2) mod \+
7=1 then S:=S+1: fi: od: S;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 54 "Por tanto hay 286 n\372meros verificando \+
dicha propiedad." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 176 "2.- Determina el conjunto de todas las soluciones del si
guiente sistema de congruencias: x congruente con 2 m\363dulo 5; x con
gruente con 3 m\363dulo 7; x congruente con 4 m\363dulo 11." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "chrem([2,3,4],[5,7,11]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "367 mod 5; 367 mod 7; 367 mo
d 11;" }}}{PARA 0 "" 0 "" {TEXT -1 92 "Por tanto cualquier soluci\363n
 es de la forma 367+5*7*11k, donde k es cualquier n\372mero entero." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "3.- \277
Cu\341ntos n\372meros entre 5 y 1000 (ambos inclusive) son primos?" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "S:=0: for i from 5 to 1000 d
o\nif isprime(i) then S:=S+1: fi: od: S;" }}}{PARA 0 "" 0 "" {TEXT -1 
164 "Hay entonces 166 primos. Como el 2 es primo y el 3 tambi\351n, es
to indica que el primo m\341s pr\363ximo a 1000 y menor que \351l ocup
a el lugar 168 en la lista de los primos:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "ithprime(168); prevprime(1000);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "4.- Sea la matriz M que \+
sigue la matriz de adyacencias de un grafo simple. Determinar el n\372
mero de v\351rtices de grado impar que tiene dicho grafo:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7626 "with(linalg): M:=matrix([[0, 1, 1
, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0,
 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, \+
1], [1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1,
 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, \+
0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1,
 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, \+
0, 1, 0, 1, 1, 0, 1, 1, 0], [0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1,
 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, \+
1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1], [0, 1, 0, 1, 0, 0, 1, 1, 0, 0,
 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, \+
0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1], [0, 0, 1, 0, 0, 0,
 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, \+
1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0,
 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, \+
1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0
, 1], [1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, \+
1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1
, 1, 1, 0, 1, 1], [1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, \+
0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0
, 0, 1, 1, 0, 1, 0, 0, 1, 1], [0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, \+
1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1
, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1], [1, 1, 0, 0, 0, 0, 1, 1, 1, \+
1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1
, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1], [0, 1, 1, 0, 0, \+
1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1
, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1], [1, \+
0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0
, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0,
 0, 0], [0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0
, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1,
 1, 1, 1, 0, 1, 0], [1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1
, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0,
 1, 0, 1, 1, 1, 0, 1, 0, 0, 1], [1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1
, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1,
 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 0, 0, 1
, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0,
 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0], [0, 0, 1, 1, 1
, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1,
 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1], [1
, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0,
 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, \+
0, 1, 0], [0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0,
 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, \+
1, 1, 1, 0, 1, 0, 1], [0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1,
 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, \+
1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0], [0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1,
 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, \+
0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1], [1, 0, 0, 0, 0, 1, 0, 0,
 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, \+
0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0], [1, 1, 1, 0,
 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, \+
1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1], \+
[0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, \+
0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1
, 1, 0, 1], [0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, \+
1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1
, 1, 1, 1, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, \+
0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0
, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0], [0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, \+
1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1
, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1], [0, 1, 1, 1, 1, 0, 1, \+
0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0
, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0], [1, 0, 0, \+
1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1
, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0]
, [1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1
, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0,
 1, 1, 1, 1], [1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0
, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1,
 0, 0, 1, 0, 1, 1, 1, 1], [1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0
, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1,
 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0], [1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0
, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0,
 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1], [0, 0, 1, 0, 1, 1, 1
, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1,
 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0], [0, 0, 1
, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1,
 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, \+
1], [0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0,
 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, \+
0, 1, 1, 1, 0], [1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1,
 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, \+
0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1,
 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, \+
1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0, 1, 1, 0,
 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, \+
1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1], [0, 1, 0, 0, 0, 1,
 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, \+
1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1], [0, 1,
 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, \+
0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0
, 1], [1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, \+
0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0
, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, \+
1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0
, 0, 1, 0, 1, 0, 0, 1, 0, 0], [1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, \+
1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0
, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1, 1, 1, \+
1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0
, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0], [0, 1, 0, 1, 0, \+
0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1
, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0], [1, \+
0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1
, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0,
 1, 1], [0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1
, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0,
 0, 1, 0, 1, 0, 0], [1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0
, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1,
 1, 1, 0, 0, 0, 0, 0, 1, 0, 0]]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "rowdim(M);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
182 "grados:=[seq(0,i=1..50)]: \nfor i from 1 to 50 do\nfor j from 1 t
o 50 do\ngrados[i]:=grados[i]+M[i,j]:\nod: od:\nS:=0:\nfor k from 1 to
 50 do \nif grados[k] mod 2<>0 then S:=S+1: fi: od: S;" }}}{PARA 0 "" 
0 "" {TEXT -1 63 "Por tanto tiene 26 v\351rtices de grado impar (y 24 \+
de grado par)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 118 "5.- Sea la siguiente matriz N la matriz que representa (
como matriz de ayacencias del digrafo asociado) una relaci\363n. " }}
{PARA 0 "" 0 "" {TEXT -1 36 "Determinar qu\351 propiedades verifica." 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "N:=matrix(6,6,[1,1,0,0,1,0
,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]);" }}}
{PARA 0 "" 0 "" {TEXT -1 60 "No es reflexiva porque la diagonal no est
\341 formada por unos." }}{PARA 0 "" 0 "" {TEXT -1 54 "No es sim\351tr
ica porque no es igual que su transpuesta:" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "N[1,3]; N[3,1];" }}}{PARA 0 "" 0 "" {TEXT -1 40 "No
 es antisim\351trica ya que, por ejemplo:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "N[1,2]; N[2,1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
77 "No es transitiva porque la matriz al cuadrado tiene entradas no nu
las nuevas:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalm(N^2)[1
,3];N[1,3];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 
"" {TEXT -1 38 "Examen de Maple. Tipo B. Febrero 2002." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "1.- En el conjunto \+
de los primeros mil n\372meros naturales al cubo, determinar cu\341nto
s de ellos no son congruentes con 1 m\363dulo 7." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "restart; with(numtheory):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 71 "S:=0: for i from 1 to 1000 do\nif i^(3) mod \+
7<>1 then S:=S+1: fi: od: S;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{PARA 0 "" 0 "" {TEXT -1 61 "Por tanto hay 571 n\372meros verificando
 la propiedad requerida." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 177 "2.- Determina el conjunto de todas las soluciones
 del siguiente sistema de congruencias: x congruente con 3 m\363dulo 1
3; x congruente con 3 m\363dulo 7; x congruente con 4 m\363dulo 11." }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "chrem([3,3,4],[13,7,11]);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "367 mod 13; 367 mod 7; 36
7 mod 11;" }}}{PARA 0 "" 0 "" {TEXT -1 104 "Por tanto cualquier soluci
\363n del sistema es de la forma 367+13*7*11k donde k es cualquier n
\372mero entero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "3.- \277Cu\341ntos n\372m
eros entre 65 y 10000 (ambos inclusive) son primos?" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 69 "S:=0: for i from 65 to 10000 do\nif isprime
(i) then S:=S+1: fi: od: S;" }}}{PARA 0 "" 0 "" {TEXT -1 225 "Hay ento
nces 1211 primos. Como el primo menor que 65 m\341s cercano a \351l es
 el 61 que ocupa el lugar 18 en la lista de los primos, entonces el pr
imo menor que 10000 m\341s cercano a \351l ocupa el lugar 1229 en la l
ista de los primos." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "prevp
rime(65);ithprime(18);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p
revprime(10^4); ithprime(1229);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "4.- Sea \+
la matriz M que sigue la matriz de adyacencias de un grafo simple. Det
erminar el n\372mero de v\351rtices de grado mayor o igual que 20 que \+
tiene dicho grafo:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7626 "with
(linalg): M:=matrix([[0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, \+
1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0
, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1], [1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, \+
0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0
, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1, 0, \+
0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0
, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0], [0, 0, 0, 0, \+
1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1
, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1], [
0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1
, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0,
 0, 1, 1], [0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1
, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0,
 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1
, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0,
 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1], [1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1
, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0,
 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1], [1, 1, 0, 1, 0, 0, 1, 0
, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0,
 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1], [0, 1, 0, 1
, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1,
 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1],
 [1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0,
 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, \+
0, 1, 0, 1], [0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0,
 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, \+
0, 1, 1, 1, 0, 0, 1, 1], [1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0,
 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, \+
0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1,
 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, \+
0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0], [1, 0, 1, 0, 0, 0, 0,
 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, \+
0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1], [1, 1, 0,
 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, \+
0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0
], [1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, \+
1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0
, 0, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, \+
0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1
, 0, 1, 0, 0, 0, 1, 0, 1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, \+
1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1
, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0], [0, 1, 1, 1, 0, 1, 1, 0, 1, 0, \+
1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0
, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1], [0, 0, 1, 1, 0, 0, \+
1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0
, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0], [0, 1, \+
1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0
, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1,
 1], [1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1
, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0,
 1, 0, 0, 1, 0], [1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0
, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0,
 0, 0, 1, 1, 1, 1, 1, 0, 1], [0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1
, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0,
 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1], [0, 0, 0, 0, 0, 1, 1, 0, 1, 1
, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1
, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1,
 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0], [0, 0
, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0,
 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, \+
0, 1], [0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0,
 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, \+
1, 0, 1, 0, 0, 0], [1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1,
 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, \+
1, 1, 1, 1, 0, 0, 0, 1, 1, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0,
 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, \+
1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1], [1, 0, 0, 0, 1, 1, 0, 1, 1,
 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, \+
0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1], [1, 1, 0, 1, 1,
 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, \+
1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0], [1,
 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, \+
1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0
, 0, 1], [0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, \+
1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0
, 0, 1, 0, 1, 0, 0], [0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, \+
1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0
, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, \+
0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0
, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0], [1, 1, 0, 1, 1, 1, 0, 1, \+
0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0
, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, \+
1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0
, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0], [
0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0
, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0,
 0, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1
, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,
 0, 0, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1
, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1,
 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0
, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1,
 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0
, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0,
 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0], [1, 1, 1, 0
, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0,
 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0],
 [0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1,
 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, \+
1, 1, 1, 0], [0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1,
 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, \+
0, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0,
 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, \+
1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0,
 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, \+
0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0], [1, 0, 0, 1, 1, 0, 1,
 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, \+
1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0]]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rowdim(M);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 177 "grados:=[seq(0,i=1..50)]: \nfor i from 1
 to 50 do\nfor j from 1 to 50 do\ngrados[i]:=grados[i]+M[i,j]:\nod: od
:\nS:=0:\nfor k from 1 to 50 do \nif grados[k] >19 then S:=S+1: fi: od
: S;" }}}{PARA 0 "" 0 "" {TEXT -1 58 "Por tanto tiene 49 v\351rtices d
e grado mayor o igual que 20." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "5.- Sea \+
la siguiente matriz N la matriz que representa (como matriz de ayacenc
ias del digrafo asociado) una relaci\363n. " }}{PARA 0 "" 0 "" {TEXT 
-1 36 "Determinar qu\351 propiedades verifica." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 89 "N:=matrix(6,6,[1,1,0,0,1,0,1,0,0,0,0,1,1,1,1,0,0
,0,0,1,0,0,1,1,1,1,1,1,1,0,0,0,0,1,1,1]);" }}}{PARA 0 "" 0 "" {TEXT 
-1 60 "No es reflexiva porque la diagonal no est\341 formada por unos.
" }}{PARA 0 "" 0 "" {TEXT -1 54 "No es sim\351trica porque no es igual
 que su transpuesta:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "N[1,
3]; N[3,1];" }}}{PARA 0 "" 0 "" {TEXT -1 40 "No es antisim\351trica ya
 que, por ejemplo:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "N[1,2]
; N[2,1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "No es transitiva por
que la matriz al cuadrado tiene entradas no nulas nuevas:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalm(N^2)[1,3];N[1,3];" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Examen
 de Maple. Tipo C. Febrero 2002." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 164 "1.- En el conjunto de los primeros 4000 \+
n\372meros naturales al cuadrado, determinar cu\341ntos de ellos son s
imult\341neamente congruentes con 1 m\363dulo 7 y con 4 m\363dulo 11.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart; with(numtheory)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "S:=0: for i from 1 to \+
4000 do\nif i^2 mod 7=1 and i^2 mod 11=4 then S:=S+1: fi: od: S;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 52 "Po
r tanto hay 52 n\372meros con la propiedad requerida." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "2.- Cu\341ntas palabr
as distintas se pueden formar permutando todas las letras de la palabr
a ornitorrinco." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(comb
inat):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "numbperm([o,r,n,i
,t,o,r,r,i,n,c,o]);" }}}{PARA 0 "" 0 "" {TEXT -1 56 "Por tanto hay exa
ctamente 3326400 palabras de este tipo." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 113 "3.- \277Cu\341ntos n\372meros entre
 1000 y 9000 (ambos inclusive) son a la vez el cuadrado y el cubo de u
n n\372mero natural?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eval
f(sqrt(1000));evalf(sqrt(9000));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "A:=\{seq(a^2,a=31..94)\}:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "evalf(9000^(1/3));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "B:=\{seq(a^3,a=10..20)\}:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "A intersect B;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "64^2; 16^3;" }}}{PARA 0 "" 0 "" {TEXT -1 56 "S\363lam
ente un n\372mero, el 4096, verifica esas condiciones." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 151 "4.- Sea la matriz M que sigue la matriz de adyacencias d
e un grafo simple. Sabiendo que dicho grafo es conexo determinar si es
 o no un grafo euleriano." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
7626 "with(linalg): M:=matrix([[0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1,
 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, \+
0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1], [1, 0, 0, 0, 1, 0, 0, 1, 1,
 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, \+
1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0,
 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, \+
1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0], [0,
 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, \+
0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1
, 1, 1], [0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, \+
0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1
, 0, 0, 0, 0, 1, 1], [0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, \+
1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0
, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, \+
1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0
, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1], [1, 1, 0, 1, 1, 1, 1, 0, \+
0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1
, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1], [1, 1, 0, 1, \+
0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1
, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1], [
0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0
, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0,
 1, 1, 1], [1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1
, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1,
 0, 1, 1, 0, 1, 0, 1], [0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0
, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0,
 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1], [1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1
, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1,
 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1, 0, 0
, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0,
 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0], [1, 0, 1, 0
, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0,
 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1],
 [1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1,
 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, \+
0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1,
 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, \+
1, 1, 0, 0, 0, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1,
 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, \+
1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0,
 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, \+
1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0], [0, 1, 1, 1, 0, 1, 1,
 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, \+
1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1], [0, 0, 1,
 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, \+
0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0
], [0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, \+
1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1
, 1, 0, 1, 1], [1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, \+
1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1
, 1, 1, 0, 1, 0, 0, 1, 0], [1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, \+
1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1
, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1], [0, 1, 0, 0, 0, 0, 0, 1, 0, 0, \+
0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1
, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1], [0, 0, 0, 0, 0, 1, \+
1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0
, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0], [0, 1, \+
0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0
, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1,
 0], [0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0
, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1,
 0, 0, 1, 0, 1], [0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0
, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 1, 1, 0, 1, 0, 0, 0], [1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1
, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1,
 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 1
, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1,
 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1], [1, 0, 0, 0, 1, 1
, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0,
 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1], [1, 1
, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1,
 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, \+
1, 0], [1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1,
 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, \+
1, 1, 0, 0, 0, 1], [0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1,
 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, \+
0, 1, 1, 0, 0, 1, 0, 1, 0, 0], [0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0,
 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, \+
0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 1, 1, 1, 0, 1,
 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, \+
1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0], [1, 1, 0, 1, 1,
 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, \+
0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0,
 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, \+
1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1
, 0, 0], [0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, \+
1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1
, 0, 0, 0, 0, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, \+
1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1
, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, \+
1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0
, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 0, 1, 0, \+
1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0
, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, \+
1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0
, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0], [
1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1
, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1,
 0, 0, 0], [0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1
, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0,
 0, 0, 0, 1, 1, 1, 0], [0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0
, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1,
 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0
, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1,
 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0, 1
, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1,
 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0], [1, 0, 0, 1
, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0,
 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0]]
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rowdim(M);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "grados:=[seq(0,i=1..50)]: \+
\nfor i from 1 to 50 do\nfor j from 1 to 50 do\ngrados[i]:=grados[i]+M
[i,j]:\nod: od:\nS:=0:\nfor k from 1 to 50 do \nif grados[k] mod 2 <>0
 then S:=S+1: fi: od: S;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 82 "Por tanto tiene v\351rtices de grado
 impar lo que implica que no puede ser euleriano." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 118 "5.- Sea la siguiente matriz N la matriz que representa (como m
atriz de ayacencias del digrafo asociado) una relaci\363n. " }}{PARA 
0 "" 0 "" {TEXT -1 36 "Determinar qu\351 propiedades verifica." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "N:=matrix(6,6,[1,1,0,0,1,0,1
,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0]);" }}}
{PARA 0 "" 0 "" {TEXT -1 57 "Es reflexiva porque la diagonal no est
\341 formada por unos." }}{PARA 0 "" 0 "" {TEXT -1 54 "No es sim\351tr
ica porque no es igual que su transpuesta:" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "N[1,3]; N[3,1];" }}}{PARA 0 "" 0 "" {TEXT -1 40 "No
 es antisim\351trica ya que, por ejemplo:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "N[1,2]; N[2,1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
77 "No es transitiva porque la matriz al cuadrado tiene entradas no nu
las nuevas:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalm(N^2)[1
,3];N[1,3];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 
"" {TEXT -1 38 "Examen de Maple. Tipo D. Febrero 2002." }}{PARA 0 "" 
0 "" {TEXT -1 134 "1.- En el conjunto de los primeros mil n\372meros n
aturales, determinar cu\341ntos de ellos son congruentes con 1 m\363du
lo 7 o con 2 m\363dulo 5." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"restart; with(numtheory):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
80 "S:=0: for i from 1 to 1000 do\nif i mod 7=1  or i mod 5=2 then S:=
S+1: fi: od: S;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "2.- Cu\341ntas pala
bras distintas empezando por la letra o se pueden formar permutando to
das las letras de la palabra ornitorrinco." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "with(combinat):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "numbperm([r,n,i,t,o,r,r,i,n,c,o]);" }}}{PARA 0 "" 0 "
" {TEXT -1 197 "Por tanto hay exactamente 831600 palabras de este tipo
. Basta observar que se selecciona una de las oes, se pone en primer l
ugar y las palabras se forman permutando entre s\355 el resto de las l
etras." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
114 "3.- \277Cu\341ntos n\372meros entre 5000 y 10000 (ambos inclusive
) son a la vez el cuadrado y el cubo de un n\372mero natural?" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf(sqrt(5000));evalf(sqrt
(15000));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A:=\{seq(a^2,a
=70..122)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalf(5000^
(1/3)); evalf(15000^(1/3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "B:=\{seq(a^3,a=17..24)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "A intersect B;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{PARA 0 "" 0 "" {TEXT -1 44 "No hay nig\372n que verifique esas condi
ciones." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 172 "4.- Sea la matriz M que sigue la \+
matriz de adyacencias de un grafo simple. Determinar el  grado del v
\351rtice 15 y el del v\351rtice 27 en la ordenaci\363n que establece \+
la matriz:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7626 "with(linalg)
: M:=matrix([[0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1,
 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, \+
1, 0, 1, 0, 0, 1, 0, 1], [1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0,
 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, \+
1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0,
 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, \+
1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0], [0, 0, 0, 0, 1, 0, 1,
 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, \+
0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1], [0, 1, 0,
 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, \+
0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1
], [0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, \+
1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0
, 0, 0, 0, 0], [1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, \+
1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0
, 1, 0, 1, 1, 1, 0, 0, 1], [1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, \+
0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1
, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1], [1, 1, 0, 1, 0, 0, 1, 0, 0, 0, \+
1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0
, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1], [0, 1, 0, 1, 0, 0, \+
0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0
, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1], [1, 1, \+
0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1
, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0,
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, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1,
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, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1,
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, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1,
 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0], [1, 0, 1, 0, 0, 0, 0, 0, 1, 0
, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0,
 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1], [1, 1, 0, 1, 0, 0
, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1,
 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0], [1, 1
, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1,
 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, \+
0, 0], [0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0,
 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, \+
0, 0, 0, 1, 0, 1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1,
 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, \+
1, 0, 0, 0, 0, 1, 1, 0, 1, 0], [0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0,
 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, \+
1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1], [0, 0, 1, 1, 0, 0, 1, 0, 0,
 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, \+
0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0], [0, 1, 1, 1, 0,
 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, \+
1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1], [1,
 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, \+
0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0
, 1, 0], [1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, \+
1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1
, 1, 1, 1, 1, 0, 1], [0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, \+
1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1
, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1], [0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, \+
0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1
, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1, 1, 1, \+
0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0
, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0], [0, 0, 0, 1, \+
1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0
, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1], [
0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1
, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1,
 0, 0, 0], [1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1
, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
 1, 0, 0, 0, 1, 1, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1
, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1,
 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1], [1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1
, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1,
 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1], [1, 1, 0, 1, 1, 1, 0, 1
, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0], [1, 0, 1, 0
, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1,
 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1],
 [0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1,
 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, \+
0, 1, 0, 0], [0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0,
 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, \+
0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0,
 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, \+
0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0], [1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1,
 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, \+
1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 0,
 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, \+
1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0], [0, 0, 1,
 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, \+
1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1
], [0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, \+
0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0
, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, \+
0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0
, 0, 0, 1, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, \+
1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0
, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0, 1, 1, \+
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0
, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0], [1, 1, 1, 0, 0, 1, \+
1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0
, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, \+
1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1
, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1,
 0], [0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1
, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1,
 1, 0, 0, 0, 0], [1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1
, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0,
 1, 0, 1, 0, 1, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1
, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1,
 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0], [1, 0, 0, 1, 1, 0, 1, 1, 1, 1
, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0,
 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0]]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rowdim(M);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 136 "grado[15]:=0: grado[17]:=0:\nfor j from 1 to 50
 do\ngrado[15]:=grado[15]+M[15,j]:\ngrado[17]:=grado[15]+M[15,j]:\nod:
 \ngrado[15]; grado[17];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 47 "Por tanto tienen grado 20 y 21 respe
ctivamente." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "5.- Sea la siguiente matriz N \+
la matriz que representa (como matriz de ayacencias del digrafo asocia
do) una relaci\363n. " }}{PARA 0 "" 0 "" {TEXT -1 36 "Determinar qu
\351 propiedades verifica." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
89 "N:=matrix(6,6,[1,1,0,0,1,0,1,1,0,0,0,1,1,1,1,0,0,1,1,1,0,1,1,1,1,1
,1,1,1,0,0,1,1,1,1,1]);" }}}{PARA 0 "" 0 "" {TEXT -1 57 "Es reflexiva \+
porque la diagonal no est\341 formada por unos." }}{PARA 0 "" 0 "" 
{TEXT -1 54 "No es sim\351trica porque no es igual que su transpuesta:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "N[1,3]; N[3,1];" }}}
{PARA 0 "" 0 "" {TEXT -1 40 "No es antisim\351trica ya que, por ejempl
o:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "N[1,2]; N[2,1];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "No es transitiva porque la matriz \+
al cuadrado tiene entradas no nulas nuevas:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "evalm(N^2)[1,3];N[1,3];" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 38 "Examen de Maple. Tipo A. Febrero 2003." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 261 12 "Ejercicio 1:" }{TEXT -1 6 "  Sea " }
{TEXT 264 7 "G=(V,E)" }{TEXT -1 29 " un multigrafo no dirigido y " }
{TEXT 262 1 "A" }{TEXT -1 38 " una matriz de adyacencias asociada a " 
}{TEXT 265 1 "G" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7612 "A:=
matrix([[0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0
, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1,
 0, 1, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1
, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,
 1, 1, 1, 0, 1, 0, 0, 1, 1, 1], [0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1
, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1,
 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1], [0, 1, 0, 0, 1, 0, 0, 0, 1
, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0,
 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1], [1, 0, 1, 1, 0
, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0,
 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1], [1
, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0,
 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, \+
1, 1, 0], [1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1,
 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, \+
1, 1, 0, 1, 0, 1, 1], [1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0,
 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, \+
0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0], [0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0,
 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, \+
0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 1, 1, 1,
 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, \+
1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1], [1, 0, 1, 1,
 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, \+
0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1], \+
[0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, \+
1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0
, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, \+
1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0
, 1, 0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, \+
0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1
, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1], [0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, \+
1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1
, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0], [1, 1, 1, 0, 1, 1, 1, \+
0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1
, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1], [1, 1, 1, \+
1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0
, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0]
, [0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1
, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1,
 1, 1, 1, 0], [1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0
, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0,
 1, 0, 0, 1, 1, 1, 1, 0], [1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0
, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1,
 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0], [0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0
, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1,
 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1], [1, 0, 1, 0, 0, 0, 0
, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1,
 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0], [1, 0, 0
, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0,
 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, \+
1], [0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0,
 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, \+
0, 1, 0, 0, 1], [0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1,
 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, \+
1, 0, 0, 0, 1, 1, 0, 1, 1], [1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0,
 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, \+
1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1], [0, 1, 0, 0, 1, 0, 0, 1, 1, 1,
 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \+
0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 1,
 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, \+
0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0], [0, 1,
 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, \+
1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1
, 1], [0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, \+
1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1
, 1, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, \+
1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1
, 1, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, \+
0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1
, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0], [0, 1, 0, 1, 1, 1, 1, 0, 0, \+
0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0
, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0], [0, 0, 0, 0, 1, \+
1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0
, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1], [0, \+
0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0
, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0,
 0, 0], [0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1
, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1,
 0, 1, 0, 0, 1, 1], [1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0
, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1,
 0, 0, 1, 0, 0, 0, 1, 0, 1, 1], [0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0
, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1,
 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 0, 0, 1, 1, 0
, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0,
 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 1
, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1,
 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1], [1
, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,
 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, \+
0, 1, 0], [1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0,
 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, \+
0, 0, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0,
 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, \+
1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0], [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0,
 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, \+
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 1, 1, 1, 1, 1,
 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, \+
1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1], [1, 0, 0, 0,
 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, \+
1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1], \+
[0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, \+
1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0
, 0, 1, 0], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, \+
1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1
, 1, 1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, \+
1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0
, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1], [0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, \+
0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0
, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0]]):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
263 "" 0 "" {TEXT 276 34 "Determinar cu\341ntos v\351rtices tiene " }
{TEXT 268 1 "G" }{TEXT 267 59 " y cu\341ntos de ellos son adyacentes a
 m\341s de 25 v\351rtices de " }{TEXT 270 2 "G." }}{PARA 0 "" 0 "" 
{TEXT 269 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rowdim(A);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "s:=0:\nfor i from 1 to
 rowdim(A) do\nd[i]:=0:\nfor j from 1 to rowdim(A) do \nd[i]:=d[i]+A[i
,j] od:\nif d[i]>25 then s:=s+1 fi:\nod:\ns;" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 12 "Ejercicio 2." }{TEXT -1 22 " S
upongamos ahora que " }{TEXT 266 2 "A " }{TEXT -1 37 "es la matriz aso
ciada a una relaci\363n " }{TEXT 273 2 "R " }{TEXT -1 18 "sobre un con
junto " }{TEXT 271 1 "B" }{TEXT -1 39 ". Determinar el n\372mero de el
ementos de " }{TEXT 272 2 "B " }{TEXT -1 82 "y las propiedades (reflex
iva, sim\351trica, antisim\351trica y transitiva) que verifica " }
{TEXT 274 2 "R." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 12 "El conjunto " }{TEXT 275 1 "B" }{TEXT -1 
20 " tiene 50 elementos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rowdim(A);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "La relaci\363n no es refl
exiva:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "A[1,1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 46 "La relaci\363n es sim\351trica pues lo es la matri
z:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "iszero(evalm(A-transpose(A)));" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "No es antisim\351trica:" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "A[1,2];\nA[2,1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 40 "Cargamos unos procedimientos necesarios:" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Matr
izdeunosyceros:=proc(M::matrix)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
  local i,j,Mat;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "  Mat:=matrix(r
owdim(M),coldim(M),0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  for i f
rom 1 to rowdim(M) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "   for j f
rom 1 to coldim(M) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "    if(M[i
,j]<>0) then Mat[i,j]:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "    fi;
 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   od;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "  od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RETURN(eva
l(Mat));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 243 "Mtrans:=proc(M::matrix)\nlocal N,S,i,j,r:
\nN:=evalm(M^2):\nS:=0:\nfor i from 1 to rowdim(M) while S=0 do \nfor \+
j from 1 to rowdim(M) do\nif N[i,j]<>0 and M[i,j]=0 then\nS:=1: fi: od
: od:\nif S=1 then r:=\"no transitiva\" else r:=\"transitiva\":\nfi:r;
 end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Mtrans(A);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 277 13 "Ejercicio 3. " 
}{TEXT -1 81 "Determinar las 10 primeras cifras del producto de los 50
 primeros n\372meros primos." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numtheory):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "p:=1:\nfor i from 1 to 50\ndo p:=p*
ithprime(i): od:\np;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 12 "Ejercic
io 4." }{TEXT -1 59 " Sea un experimento de Bernoulli con probabilidad
 de \351xito " }{TEXT 279 9 "p=0.001. " }{TEXT -1 137 "Determinar el m
\355nimo n\372mero de repeticiones que han de efectuarse para que la p
robabilidad de obtener al menos 1 \351xito sea mayor que 0.5." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "with(combinat):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=n
->1-(1-0.001)^n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "i:=1: \nwhile f(i)<=0.5 do
\ni:=i+1 od: i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f(692);\nf(693);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "
Examen de Maple. Tipo B. Febrero 2003." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 280 12 "Ejercicio 1:" }{TEXT -1 6 "  Sea " }{TEXT 283 7 "G=(V,E)
" }{TEXT -1 29 " un multigrafo no dirigido y " }{TEXT 281 1 "M" }
{TEXT -1 38 " una matriz de adyacencias asociada a " }{TEXT 284 1 "G" 
}{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "restart; with(linalg):" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7612 "M:=matrix(
[[1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1,
 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, \+
0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0,
 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, \+
1, 0, 1, 0, 0, 1, 1, 1], [0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1,
 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, \+
0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1], [0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1,
 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, \+
0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1], [1, 0, 1, 1, 0, 0, 1,
 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, \+
0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1], [1, 0, 0,
 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, \+
0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0
], [1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, \+
1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0
, 1, 0, 1, 1], [1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, \+
0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0
, 1, 0, 1, 0, 1, 1, 0, 0], [0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, \+
0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1
, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 1, 1, 1, 0, 0, \+
0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1
, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1], [1, 0, 1, 1, 1, 1, \+
1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0
, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1], [0, 0, \+
0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1
, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0,
 0], [0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1
, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0,
 0, 0, 1, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1
, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1,
 1, 0, 0, 1, 1, 0, 1, 0, 1], [0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1
, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1,
 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0], [1, 1, 1, 0, 1, 1, 1, 0, 0, 0
, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0,
 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1], [1, 1, 1, 1, 0, 1
, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0,
 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0], [0, 0
, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1,
 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, \+
1, 0], [1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1,
 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, \+
0, 1, 1, 1, 1, 0], [1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0,
 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, \+
1, 0, 0, 1, 1, 0, 1, 1, 1, 0], [0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1,
 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, \+
1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1], [1, 0, 1, 0, 0, 0, 0, 1, 0,
 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, \+
0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0], [1, 0, 0, 1, 1,
 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, \+
0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1], [0,
 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, \+
1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0
, 0, 1], [0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, \+
1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0
, 0, 1, 1, 0, 1, 1], [1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, \+
0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0
, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1], [0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, \+
0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 1, 1, 0, \+
0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0
, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0], [0, 1, 1, 0, \+
1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1
, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1], [
0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0
, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0,
 1, 0, 0], [1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1
, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1,
 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1
, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0,
 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0], [0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0
, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0,
 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0
, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1,
 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1], [0, 0, 0, 0
, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0,
 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0],
 [0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1,
 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, \+
0, 0, 1, 1], [1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1,
 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, \+
1, 0, 0, 0, 1, 0, 1, 1], [0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0,
 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, \+
1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0,
 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, \+
1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 1, 0, 0,
 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, \+
1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1], [1, 1, 0,
 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, \+
0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0
], [1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, \+
1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1
, 1, 0, 0, 0], [0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, \+
1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0
, 0, 0, 0, 0, 1, 1, 1, 0], [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, \+
0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 1, 1, 1, 1, 1, 0, 0, \+
0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1
, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1], [1, 0, 0, 0, 0, 1, \+
0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1
, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1], [0, 0, \+
0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1
, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1,
 0], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0
, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1,
 1, 0, 0, 0, 1], [0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1
, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1,
 0, 1, 0, 0, 0, 1, 0, 0, 1], [0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1
, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1,
 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0]]):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 
"" {TEXT 294 34 "Determinar cu\341ntos v\351rtices tiene " }{TEXT 287 
1 "G" }{TEXT 286 87 ". Determinar los v\351rtices que son adyacentes c
onsigo mismos, esto es, cuantos v\351rtices " }{TEXT 298 1 "a" }{TEXT 
299 15 " son tales que " }{TEXT 300 6 "(a,a) " }{TEXT 301 104 "es una \+
arista. De entre los v\351rtices no adyacentes consigo mismos, determi
nar los de grado mayor que 30." }}{PARA 0 "" 0 "" {TEXT 288 1 " " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rowdim(M);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "seq(M[i,i],i=1..50);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 129 "s:=0:\nfor i from 4 to rowdim(M) do\nd[i]:=
0:\nfor j from 1 to rowdim(M) do \nd[i]:=d[i]+M[i,j] od:\nif d[i]>30 t
hen s:=s+1 fi:\nod:\ns;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 282 12 "Ejercicio 2." }{TEXT -1 22 " Supongamos ahora que " }
{TEXT 285 2 "M " }{TEXT -1 37 "es la matriz asociada a una relaci\363n
 " }{TEXT 291 2 "R " }{TEXT -1 18 "sobre un conjunto " }{TEXT 289 1 "B
" }{TEXT -1 39 ". Determinar el n\372mero de elementos de " }{TEXT 
290 2 "B " }{TEXT -1 31 "y las propiedades que verifica " }{TEXT 292 
2 "R." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 12 "El conjunto " }{TEXT 293 1 "B" }{TEXT -1 20 " tiene 50 \+
elementos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "rowdim(M);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 28 "La relaci\363n no es reflexiva:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "M[4
,4];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "
La relaci\363n es sim\351trica pues lo es la matriz:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "iszero(eva
lm(M-transpose(M)));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 20 "No es antisim\351trica:" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "M[1,2];\nM[2,1];" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Cargamos \+
unos procedimientos necesarios:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Matrizdeunosyceros:=proc(M::
matrix)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "  local i,j,Mat;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "  Mat:=matrix(rowdim(M),coldim(M),0
);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  for i from 1 to rowdim(M) d
o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "   for j from 1 to coldim(M) d
o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "    if(M[i,j]<>0) then Mat[i,j
]:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "    fi; " }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 6 "   od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RETURN(eval(Mat));" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
243 "Mtrans:=proc(M::matrix)\nlocal N,S,i,j,r:\nN:=evalm(M^2):\nS:=0:
\nfor i from 1 to rowdim(M) while S=0 do \nfor j from 1 to rowdim(M) d
o\nif N[i,j]<>0 and M[i,j]=0 then\nS:=1: fi: od: od:\nif S=1 then r:=
\"no transitiva\" else r:=\"transitiva\":\nfi:r; end:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Mtrans(M);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 295 13 "Ejercicio 3. " }{TEXT -1 69 "Determi
nar las 10 primeras cifras del producto de los n\372meros primos " }
{TEXT 302 2 "p " }{TEXT -1 10 "tales que " }{TEXT 303 11 "1000<p<2000
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "with(numtheory):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "p:=1:\nfor i from 1001 to 1999\ndo if isprime(i) then
 p:=p*i: fi: od:\np;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 12 "Ejercic
io 4." }{TEXT -1 59 " Sea un experimento de Bernoulli con probabilidad
 de \351xito " }{TEXT 297 8 "p=0.01. " }{TEXT -1 138 "Determinar el m
\355nimo n\372mero de repeticiones que han de efectuarse para que la p
robabilidad de obtener al menos 1 \351xito sea mayor que 0.75." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "with(combinat):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=n
->1-(1-0.01)^n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "i:=1: \nwhile f(i)<=0.75 do
\ni:=i+1 od: i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f(138);\nf(137);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "
Examen de Maple. Tipo C. Febrero 2003." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 316 12 "Ejercicio 1:" }{TEXT -1 6 "  Sean" }
{TEXT 310 8 " G=(V,E)" }{TEXT -1 25 "  un grafo no dirigido y " }
{TEXT 304 1 "A" }{TEXT -1 64 ", definida a continuaci\363n, una matriz
 de adyacencias asociada a " }{TEXT 317 1 "G" }{TEXT -1 1 ":" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wi
th(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7022 "A := matri
x([[0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, \+
1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0
, 0, 1], [0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, \+
0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1
, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, \+
1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1
, 0, 0, 0, 0, 0, 1], [0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, \+
0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0
, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, \+
1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1
, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0], [1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, \+
0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0
, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1], [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, \+
0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0
, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1], [0, 1, 1, 0, 1, 0, 1, 0, \+
0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1
, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0], [1, 0, 0, 1, 0, 1, \+
0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1
, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0], [0, 0, 0, 1, \+
0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0
, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0], [1, 1, \+
0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1
, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0], [
0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0
, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0,
 1], [0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1
, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1,
 1, 1, 0], [0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1
, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0,
 1, 1, 1, 1, 0], [0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1
, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0,
 0, 0, 0, 1, 0, 1, 0], [1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0
, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0,
 0, 1, 1, 0, 0, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1
, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0,
 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1], [0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0
, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0,
 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0], [1, 1, 1, 1, 1, 1, 0, 1, 1, 0
, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0,
 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0], [1, 0, 1, 1, 0, 0, 0, 0
, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1,
 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0], [1, 0, 1, 0, 0, 0
, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1,
 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], [1, 1, 0, 1
, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1,
 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1], [1, 0
, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1,
 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1],
 [0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1,
 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, \+
1, 0], [0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1,
 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, \+
1, 0, 1, 1], [1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1,
 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, \+
1, 0, 0, 1, 1, 1], [0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0,
 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, \+
1, 0, 1, 0, 1, 0, 1, 1], [1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0,
 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, \+
1, 1, 1, 0, 1, 0, 0, 0, 0, 1], [1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1,
 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, \+
1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0,
 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, \+
1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1], [1, 1, 0, 1, 1, 1, 0, 1, 1,
 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, \+
0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 1,
 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, \+
0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 0, 1,
 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, \+
0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1], [1, 0, 1,
 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, \+
1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0,
 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, \+
1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0
], [0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, \+
0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1
, 0, 1], [0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, \+
1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1
, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, \+
1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1
, 1, 0, 1, 1, 1, 1], [0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, \+
0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0
, 1, 0, 0, 1, 1, 1, 0, 1], [1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, \+
1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1
, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0], [1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, \+
0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0
, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1], [1, 0, 1, 0, 0, 0, 1, 0, 1, 1, \+
0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0
, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1], [0, 1, 0, 1, 1, 1, 1, 1, \+
1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0
, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1], [1, 1, 0, 0, 1, 0, \+
0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0
, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1], [0, 1, 0, 1, \+
1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0
, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0], [0, 0, \+
0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1
, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0], [
0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1
, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0,
 1], [1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1
, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0,
 0, 1, 0]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Supongamos ahor
a que " }{TEXT 334 1 "A" }{TEXT -1 38 " es la matriz asociada a una re
laci\363n " }{TEXT 306 2 "R " }{TEXT -1 18 "sobre un conjunto " }
{TEXT 304 1 "B" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Determ
inar el n\372mero de elementos de " }{TEXT 330 1 "B" }{TEXT 305 1 " " 
}{TEXT -1 31 "y las propiedades que verifica " }{TEXT 333 1 "R" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 "rowdim(A);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 12 "El conjunto " }{TEXT 331 1 "B" }{TEXT -1 
20 " tiene 48 elementos." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "A[1,1]
;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "La \+
relaci\363n no es reflexiva." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "is
zero(evalm(A-transpose(A)));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 46 "La relaci\363n es sim\351trica pues lo es
 la matriz " }{TEXT 311 1 "A" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "A[1,5];\nA[5,1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 32 "La relaci\363n no es antisim\351trica." }
}{PARA 0 "" 0 "" {TEXT -1 76 "Cargamos unos procedimientos necesarios \+
para el estudio de la transitividad:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Matrizdeunosyceros:=proc(M
::matrix)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "  local i,j,Mat;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "  Mat:=matrix(rowdim(M),coldim(M),0
);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  for i from 1 to rowdim(M) d
o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "   for j from 1 to coldim(M) d
o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "    if(M[i,j]<>0) then Mat[i,j
]:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "    fi; " }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 6 "   od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RETURN(eval(Mat));" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
243 "Mtrans:=proc(M::matrix)\nlocal N,S,i,j,r:\nN:=evalm(M^2):\nS:=0:
\nfor i from 1 to rowdim(M) while S=0 do \nfor j from 1 to rowdim(M) d
o\nif N[i,j]<>0 and M[i,j]=0 then\nS:=1: fi: od: od:\nif S=1 then r:=
\"no transitiva\" else r:=\"transitiva\":\nfi:r; end:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Mtrans(A);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "La relaci\363n no es tran
sitiva." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 305 12 "Ejercicio 2." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 23 "Determinar si el grafo " }{TEXT 312 1 "G" }{TEXT -1 36 " es sim
ple. Calcular el cardinal de " }{TEXT 307 1 "E" }{TEXT -1 3 " . " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Ya que" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "iszero(Matrizdeunosyceros(A)
-A);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 1 "
 " }{TEXT -1 26 "las entradas de la matriz " }{TEXT 332 1 "A" }{TEXT 
-1 59 " son todos ceros y unos. Vimos que la matriz es sim\351trica. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Tenem
os que verificar que no hay lazos: " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "s:=0:\nfor i from 1 to rowdim(A) do\nif A[i,i]<>0 the
n s:=s+1 fi:\nod:\ns;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Hay 21
 lazos y el grafo " }{TEXT 313 1 "G" }{TEXT -1 16 "  no es simple. " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 120 "a:=0:\nfor i from 1 to rowdim(A) do\n    for j from i to coldim
(A) do\n       if A[i,j]=1 then a:=a+1: fi:\n    od:\nod: \na; " }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "El cardin
al de " }{TEXT 314 3 "E  " }{TEXT -1 15 "es igual a 593." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 308 13 "Ejercicio 3. " }{TEXT 
-1 31 "Determinar cu\341ntos v\351rtices en " }{TEXT 315 2 "V " }
{TEXT -1 40 " tienen grado congruente con 1 m\363dulo 2." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(
numtheory):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 97 "Tenemos que calcular el grado de cada v\351rtice teniendo en cu
enta los lazos, que a\361aden 2 (y no 1)" }}{PARA 0 "" 0 "" {TEXT -1 
21 "al grado del v\351rtice:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "s:=0:\nfor i from 1 to rowd
im(A) do\n    g[i]:=0:\n    for j from 1 to coldim(A) do\n        g[i]
:=g[i]+A[i,j]: \n    od:\n    g[i]:=g[i]+A[i,i]:\n    if g[i] mod 2= 1
  then s:=s+1: fi:\nod:\ns;\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 31 "Hay 16 v\351rtices de grado impar." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 12 "Ejercicio 4." }{TEXT -1 11 " P
ara todo " }{TEXT 318 1 "n" }{TEXT 326 3 ",  " }{TEXT 327 1 "n" }
{TEXT -1 16 "=1,...,48, sea  " }{XPPEDIT 18 0 "A[n];" "6#&%\"AG6#%\"nG
" }{TEXT -1 30 " una matriz obtenida borrando " }{TEXT 319 1 "n" }
{TEXT -1 30 " filas y las correspondientes " }{TEXT 320 1 "n" }{TEXT 
-1 9 " columnas" }}{PARA 0 "" 0 "" {TEXT -1 14 "de la matriz  " }
{TEXT 321 4 "A.  " }{TEXT -1 46 "Cada una de estas matrices define un \+
subgrafo " }{XPPEDIT 18 0 "G[n];" "6#&%\"GG6#%\"nG" }{TEXT -1 4 " de \+
" }{TEXT 322 1 "G" }{TEXT -1 10 "  (con 48-" }{TEXT 323 1 "n" }{TEXT 
-1 11 " v\351rtices)." }}{PARA 0 "" 0 "" {TEXT -1 11 "Sea ahora  " }
{XPPEDIT 18 0 "s[n];" "6#&%\"sG6#%\"nG" }{TEXT -1 24 " el n\372mero de
 subgrafos " }{XPPEDIT 18 0 "G[n]" "6#&%\"GG6#%\"nG" }{TEXT -1 61 " qu
e se pueden generar de esta manera para un  valor dado de " }{TEXT 
324 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "Verificar si \+
existen valores de " }{TEXT 325 2 "n " }{TEXT -1 10 "tales que " }
{XPPEDIT 18 0 "s[n]" "6#&%\"sG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 
0 "1677106640;" "6#\"+Sm5x;" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(combinat):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for n from 1 to 48 do\n   \+
s[n]:=((binomial(48,n))); od;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for i from 1 to 48 do\nif s[
i]=1677106640 then print(i) fi:\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "s[9];s[39];" }}}{PARA 0 
"" 0 "" {TEXT 328 1 "n" }{TEXT -1 5 "=9 y " }{TEXT 329 1 "n" }{TEXT 
-1 5 "= 39." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 4 "#fin" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Examen de Maple. Tipo D. Febrer
o 2003." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 347 
12 "Ejercicio 1:" }{TEXT -1 6 "  Sean" }{TEXT 341 8 " G=(V,E)" }{TEXT 
-1 25 "  un grafo no dirigido y " }{TEXT 335 1 "A" }{TEXT -1 64 ", def
inida a continuaci\363n, una matriz de adyacencias asociada a " }
{TEXT 348 1 "G" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7022 "A := matrix([[0, 0, 0, 0, 1, 1, 0
, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0,
 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0
, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0,
 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0], [0, 1, 1
, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0,
 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1], [0
, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0,
 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, \+
0], [1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1,
 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, \+
0, 0, 0], [1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0,
 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, \+
0, 0, 0, 1, 1], [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1,
 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, \+
1, 1, 0, 1, 1, 0, 1], [0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0,
 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, \+
0, 1, 0, 1, 1, 1, 1, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0,
 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, \+
0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0], [0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0,
 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, \+
0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0], [1, 1, 0, 1, 1, 1, 0, 0, 0, 1,
 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, \+
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0], [0, 1, 1, 0, 1, 0, 1, 0,
 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, \+
1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1], [0, 1, 0, 0, 1, 0,
 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, \+
0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0], [0, 1, 0, 0,
 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, \+
0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0], [0, 0,
 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, \+
1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0], \+
[1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, \+
0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0
, 1], [1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, \+
1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0
, 0, 0, 1], [0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, \+
1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1
, 0, 1, 1, 0, 0], [1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, \+
0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0
, 1, 0, 0, 1, 0, 1, 0], [1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, \+
1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0
, 0, 0, 1, 0, 1, 1, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, \+
1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1
, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], [1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, \+
1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0
, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 1, 0, 1, \+
1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0
, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1], [0, 0, 1, 0, 1, 1, 1, \+
1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1
, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0], [0, 0, 1, 0, 0, \+
0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0
, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1], [1, 1, 0, \+
1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1
, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1], [0, \+
0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0
, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1]
, [1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1
, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0,
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, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1,
 0, 0, 0, 1], [0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1
, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1,
 1, 0, 0, 0, 1, 1], [1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1
, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1,
 1, 0, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0
, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0,
 0, 1, 1, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1
, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0,
 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1], [1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0
, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0,
 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 1, 1, 0, 1, 0, 1
, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0,
 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0], [0, 1, 1, 0, 1, 0, 1
, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 1
, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0,
 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0], [0, 0, 0
, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1,
 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1], [0
, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1,
 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, \+
1], [1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0,
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, \+
0, 0, 0], [1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0,
 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, \+
1, 0, 1, 0, 1], [1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0,
 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, \+
0, 0, 0, 1, 1, 0, 1], [0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0,
 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, \+
1, 0, 0, 0, 1, 1, 0, 1, 1], [1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1,
 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, \+
0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1], [0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0,
 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, \+
0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 0, 1,
 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, \+
1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0], [0, 0, 0, 1, 0, 1, 0, 0,
 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, \+
0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1], [1, 0, 1, 0, 0, 1,
 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, \+
1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0]]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Supongamos ahora que " }{TEXT 
366 1 "A" }{TEXT -1 38 " es la matriz asociada a una relaci\363n " }
{TEXT 337 2 "R " }{TEXT -1 18 "sobre un conjunto " }{TEXT 335 1 "B" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Determinar el n\372mero
 de elementos de " }{TEXT 362 1 "B" }{TEXT 336 1 " " }{TEXT -1 82 "y l
as propiedades (reflexiva, sim\351trica, antisim\351trica y transitiva
) que verifica " }{TEXT 363 1 "R" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rowdim(A);" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "El con
junto " }{TEXT 364 1 "B" }{TEXT -1 20 " tiene 48 elementos." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "A[1,1];" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "La relaci\363n no es reflexiva.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "iszero(evalm(A-transpose(A))
);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "La
 relaci\363n es sim\351trica pues lo es la matriz " }{TEXT 342 1 "A" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A[1,5];\nA[5
,1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "
La relaci\363n no es antisim\351trica." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 76 "Cargamos unos procedimientos necesari
os para el estudio de la transitividad:" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Matrizdeunosyceros:=pro
c(M::matrix)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "  local i,j,Mat;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "  Mat:=matrix(rowdim(M),coldim(M),
0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  for i from 1 to rowdim(M) \+
do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "   for j from 1 to coldim(M) \+
do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "    if(M[i,j]<>0) then Mat[i,
j]:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "    fi; " }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 6 "   od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RETURN(eval(Mat));" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 243 "Mtrans:=proc(M::matrix)\nlocal N,S,i,j,r:\nN:=evalm(M^2):\nS:=0
:\nfor i from 1 to rowdim(M) while S=0 do \nfor j from 1 to rowdim(M) \+
do\nif N[i,j]<>0 and M[i,j]=0 then\nS:=1: fi: od: od:\nif S=1 then r:=
\"no transitiva\" else r:=\"transitiva\":\nfi:r; end:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Mtrans(A);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "La relaci\363n no es tran
sitiva." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 336 12 "Ejercicio 2." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 23 "Determinar si el grafo " }{TEXT 343 1 "G" }{TEXT -1 36 " es sim
ple. Calcular el cardinal de " }{TEXT 338 1 "E" }{TEXT -1 3 " . " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Ya que" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "iszero(Matrizdeunosyceros(A)-A);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 349 1 " " }{TEXT -1 26 "las entradas de la m
atriz " }{TEXT 365 2 "A " }{TEXT -1 58 "son todos ceros y unos. Vimos \+
que la matriz es sim\351trica. " }}{PARA 0 "" 0 "" {TEXT 336 1 " " }
{TEXT -1 40 "Tenemos que verificar que no hay lazos: " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "s:=0:\nfo
r i from 1 to rowdim(A) do\nif A[i,i]<>0 then s:=s+1 fi:\nod:\ns;" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Hay 23 lazos y el grafo \+
" }{TEXT 344 1 "G" }{TEXT -1 16 "  no es simple. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "a:=0:\nfor \+
i from 1 to rowdim(A) do\n    for j from i to coldim(A) do\n       if \+
A[i,j]=1 then a:=a+1: fi:\n    od:\nod: \na; " }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "El cardinal de " }{TEXT 
345 3 "E  " }{TEXT -1 15 "es igual a 595." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 339 13 "Ejercicio 3. " }{TEXT -1 31 "Determinar cu\341n
tos v\351rtices en " }{TEXT 346 2 "V " }{TEXT -1 40 " tienen grado con
gruente con 1 m\363dulo 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numtheory):" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Tenemos que calcular el
 grado de cada v\351rtice teniendo en cuenta los lazos, que a\361aden \+
2 (y no 1)" }}{PARA 0 "" 0 "" {TEXT -1 22 "al grado del v\351rtice :" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 187 "s:=0:\nfor i from 1 to rowdim(A) do\n    g[i]:=0:\n    for j fr
om 1 to coldim(A) do\n        g[i]:=g[i]+A[i,j]: \n    od:\n    g[i]:=
g[i]+A[i,i]:\n    if g[i] mod 2 = 1  then s:=s+1: fi:\nod:\ns;\n" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Hay  16 v
\351rtices de grado impar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 340 12 "Ejercicio 4." }{TEXT -1 11 " Para todo \+
" }{TEXT 350 1 "n" }{TEXT 358 3 ",  " }{TEXT 359 1 "n" }{TEXT -1 16 "=
1,...,48, sea  " }{XPPEDIT 18 0 "A[n];" "6#&%\"AG6#%\"nG" }{TEXT -1 
30 " una matriz obtenida borrando " }{TEXT 351 1 "n" }{TEXT -1 30 " fi
las y las correspondientes " }{TEXT 352 1 "n" }{TEXT -1 24 " columnas \+
de la matriz  " }{TEXT 353 4 "A.  " }{TEXT -1 46 "Cada una de estas ma
trices define un subgrafo " }{XPPEDIT 18 0 "G[n];" "6#&%\"GG6#%\"nG" }
{TEXT -1 4 " de " }{TEXT 354 1 "G" }{TEXT -1 9 " (con 48-" }{TEXT 355 
1 "n" }{TEXT -1 11 " v\351rtices)." }}{PARA 0 "" 0 "" {TEXT -1 11 "Sea
 ahora  " }{XPPEDIT 18 0 "s[n];" "6#&%\"sG6#%\"nG" }{TEXT -1 24 " el n
\372mero de subgrafos " }{XPPEDIT 18 0 "G[n]" "6#&%\"GG6#%\"nG" }
{TEXT -1 60 " que se pueden generar de esta manera para un valor dado \+
de " }{TEXT 356 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "V
erificar si existen valores de " }{TEXT 357 2 "n " }{TEXT -1 10 "tales
 que " }{XPPEDIT 18 0 "s[n]" "6#&%\"sG6#%\"nG" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "12271512;" "6#\")7:F7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(combi
nat):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for n from 1 to 48
 do\n   s[n]:=((binomial(48,n))); od;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "for i from 1 to 48 do\nif s[i]=12271512 then print(i)
 fi:\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "s[6];s[42];" }
}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 360 1 "n" }{TEXT -1 5 "=6 y " }{TEXT 361 1 "n" }
{TEXT -1 5 "= 42." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 4 "#fin" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "
Examen de Maple. Tipo E. Febrero 2003." }}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 56 "Examen 5 de Maple. Matem\341tica Discreta. Curso 2002/200
3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 394 13 "Ejercicio 1: " }{TEXT -1 5 " Sea " }{TEXT 411 1 "n" }
{TEXT -1 41 " un n\372mero natural, definimos la funcion " }{TEXT 406 
4 "T(n)" }{TEXT -1 67 " de la forma siguiente: si la descomposici\363n
 en factores primos de " }{TEXT 407 2 "n " }{TEXT -1 4 "es  " }{TEXT 
408 6 "n = ( " }{XPPEDIT 18 0 "p[1]^a[1];" "6#)&%\"pG6#\"\"\"&%\"aG6#
\"\"\"" }{TEXT 414 5 " )*( " }{XPPEDIT 18 0 "p[2]^a[2];" "6#)&%\"pG6#
\"\"#&%\"aG6#\"\"#" }{TEXT 412 9 ")* ...*( " }{XPPEDIT 18 0 "p[s]^a[s]
;" "6#)&%\"pG6#%\"sG&%\"aG6#F'" }{TEXT 413 2 ") " }{TEXT -1 10 " enton
ces " }}{PARA 0 "" 0 "" {TEXT 409 9 "T(n)=T( (" }{XPPEDIT 18 0 "p[1]^a
[1]" "6#)&%\"pG6#\"\"\"&%\"aG6#\"\"\"" }{TEXT 415 3 ")*(" }{XPPEDIT 
18 0 "p[2]^a[2]" "6#)&%\"pG6#\"\"#&%\"aG6#\"\"#" }{TEXT 416 7 ")*...*(
" }{XPPEDIT 18 0 "p[s]^a[s];" "6#)&%\"pG6#%\"sG&%\"aG6#F'" }{TEXT 417 
6 ") )= (" }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT 418 5 "+1)
*(" }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT 419 9 "+1)*...*(" 
}{XPPEDIT 18 0 "a[s];" "6#&%\"aG6#%\"sG" }{TEXT 420 4 "+1)." }}{PARA 
0 "" 0 "" {TEXT -1 20 "Calcula el valor de " }{TEXT 410 8 "T(48510)" }
{TEXT -1 3 ":  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "with(numtheory):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "ifactor(48510);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "T(48510)=(1+1)*(2+1)*(1+1)*(2+1)*(1+1);" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 395 13 "Ej
ercicio 2: " }{TEXT -1 39 "Determinar cu\341ntos n\372meros de la form
a:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "2^p-1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 16 "son primos, con " }{TEXT 396 1 "p" }{TEXT -1 87 " to
mando valores entre 1 y 29 (ambos inclusive). Determinar exactamente l
os valores de " }{TEXT 397 1 "p" }{TEXT -1 22 " que los hacen primos.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 112 "j:=0:\nfor i from 1 to 29 do \nif( isprime(2**i -1) ) then j:
=j+1; \nprint(i) fi ;od; \nprint('numero-de-primos',j);" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 398 13 "Ej
ercicio 3: " }{TEXT -1 83 "Determinar, si existen, todas las solucione
s del sistema de congruencias siguiente:" }}{PARA 0 "" 0 "" {TEXT -1 
11 "7x=3(mod11)" }}{PARA 0 "" 0 "" {TEXT -1 10 "5x=1(mod7)" }}{PARA 0 
"" 0 "" {TEXT -1 10 "3x=2(mod5)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "msolve(7*x=3,11);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "msolve(5*x=1,7);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "msolve(3*x=2,5);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "chrem([2,3,4],[11,7,5]); 11*7*5;" }}}
{PARA 0 "" 0 "" {TEXT -1 16 "La respuesta es " }{TEXT 399 8 "24+ 385k
" }{TEXT -1 5 " con " }{TEXT 400 1 "k" }{TEXT -1 18 " un n\372mero ent
ero." }{MPLTEXT 1 0 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 401 13 "Ejercicio 4: " }{TEXT -1 35 "El m
\351todo de cifrado conocido como " }{TEXT 402 13 "transposici\363n" }
{TEXT -1 103 " consiste en cambiar de lugar las letras de un mensaje p
ara hacerlo incomprensible. Usando la librer\355a " }{TEXT 403 8 "comb
inat" }{TEXT -1 24 " descifrar el mensaje:  " }{TEXT 404 7 "otxei. " }
{TEXT -1 58 "(Sabiendo que se ha de recibir una palabra en castellano.
)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "restart;\nwith(combinat):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "permute([o,t,x,e,i]);" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "La respuesta est\341 entre las po
sibles permutaciones del mensaje recibido, y solo tiene sentido la pal
abra " }{TEXT 405 5 "exito" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 5 "2003." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 367 13 "Ejercicio 1: " 
}{TEXT -1 5 " Sea " }{TEXT 384 1 "n" }{TEXT -1 41 " un n\372mero natur
al, definimos la funcion " }{TEXT 379 4 "T(n)" }{TEXT -1 67 " de la fo
rma siguiente: si la descomposici\363n en factores primos de " }{TEXT 
380 2 "n " }{TEXT -1 4 "es  " }{TEXT 381 6 "n = ( " }{XPPEDIT 18 0 "p[
1]^a[1];" "6#)&%\"pG6#\"\"\"&%\"aG6#\"\"\"" }{TEXT 387 5 " )*( " }
{XPPEDIT 18 0 "p[2]^a[2];" "6#)&%\"pG6#\"\"#&%\"aG6#\"\"#" }{TEXT 385 
9 ")* ...*( " }{XPPEDIT 18 0 "p[s]^a[s];" "6#)&%\"pG6#%\"sG&%\"aG6#F'
" }{TEXT 386 2 ") " }{TEXT -1 10 " entonces " }}{PARA 0 "" 0 "" {TEXT 
382 9 "T(n)=T( (" }{XPPEDIT 18 0 "p[1]^a[1]" "6#)&%\"pG6#\"\"\"&%\"aG6
#\"\"\"" }{TEXT 388 3 ")*(" }{XPPEDIT 18 0 "p[2]^a[2]" "6#)&%\"pG6#\"
\"#&%\"aG6#\"\"#" }{TEXT 389 7 ")*...*(" }{XPPEDIT 18 0 "p[s]^a[s];" "
6#)&%\"pG6#%\"sG&%\"aG6#F'" }{TEXT 390 6 ") )= (" }{XPPEDIT 18 0 "a[1]
;" "6#&%\"aG6#\"\"\"" }{TEXT 391 5 "+1)*(" }{XPPEDIT 18 0 "a[2];" "6#&
%\"aG6#\"\"#" }{TEXT 392 9 "+1)*...*(" }{XPPEDIT 18 0 "a[s];" "6#&%\"a
G6#%\"sG" }{TEXT 393 4 "+1)." }}{PARA 0 "" 0 "" {TEXT -1 20 "Calcula e
l valor de " }{TEXT 383 8 "T(48510)" }{TEXT -1 3 ":  " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numt
heory):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ifactor(48510);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "T(48510)=(1+1)*(2+1)*(1
+1)*(2+1)*(1+1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 368 13 "Ejercicio 2: " }{TEXT -1 39 "Determinar c
u\341ntos n\372meros de la forma:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "2^p-1;" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "son primos, con " }{TEXT 
369 1 "p" }{TEXT -1 87 " tomando valores entre 1 y 29 (ambos inclusive
). Determinar exactamente los valores de " }{TEXT 370 1 "p" }{TEXT -1 
22 " que los hacen primos." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "j:=0:\nfor i from 1 to 29 do \nif(
 isprime(2**i -1) ) then j:=j+1; \nprint(i) fi ;od; \nprint('numero-de
-primos',j);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 371 13 "Ejercicio 3: " }{TEXT -1 83 "Determinar, \+
si existen, todas las soluciones del sistema de congruencias siguiente
:" }}{PARA 0 "" 0 "" {TEXT -1 11 "7x=3(mod11)" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "5x=1(mod7)" }}{PARA 0 "" 0 "" {TEXT -1 10 "3x=2(mod5)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "msolve(7*x=3,11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ms
olve(5*x=1,7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "msolve(3*
x=2,5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "chrem([2,3,4],[1
1,7,5]); 11*7*5;" }}}{PARA 0 "" 0 "" {TEXT -1 16 "La respuesta es " }
{TEXT 372 8 "24+ 385k" }{TEXT -1 5 " con " }{TEXT 373 1 "k" }{TEXT -1 
18 " un n\372mero entero." }{MPLTEXT 1 0 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 374 13 "Ejercicio
 4: " }{TEXT -1 35 "El m\351todo de cifrado conocido como " }{TEXT 
375 13 "transposici\363n" }{TEXT -1 103 " consiste en cambiar de lugar
 las letras de un mensaje para hacerlo incomprensible. Usando la libre
r\355a " }{TEXT 376 8 "combinat" }{TEXT -1 24 " descifrar el mensaje: \+
 " }{TEXT 377 7 "otxei. " }{TEXT -1 58 "(Sabiendo que se ha de recibir
 una palabra en castellano.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart;\nwith(combinat):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "permute([o,t,x,e,i]);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "La respu
esta est\341 entre las posibles permutaciones del mensaje recibido, y \+
solo tiene sentido la palabra " }{TEXT 378 5 "exito" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 
"Examen de Maple. Tipo F. Febrero 2003." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 421 13 "Ejercicio 1: " }
{TEXT -1 5 " Sea " }{TEXT 433 1 "n" }{TEXT -1 31 " un n\372mero natura
l. La funci\363n " }{TEXT 434 8 "sigma(n)" }{TEXT -1 82 " se define se
g\372n la f\363rmula siguiente: si la descomposici\363n en factores pr
imos de " }{TEXT 435 1 "n" }{TEXT -1 6 " es   " }{TEXT 439 6 "n = ( " 
}{XPPEDIT 18 0 "p[1]^a[1];" "6#)&%\"pG6#\"\"\"&%\"aG6#\"\"\"" }{TEXT 
442 5 " )*( " }{XPPEDIT 18 0 "p[2]^a[2];" "6#)&%\"pG6#\"\"#&%\"aG6#\"
\"#" }{TEXT 440 9 ")* ...*( " }{XPPEDIT 18 0 "p[s]^a[s];" "6#)&%\"pG6#
%\"sG&%\"aG6#F'" }{TEXT 441 4 ")   " }{TEXT -1 8 "entonces" }}{PARA 0 
"" 0 "" {TEXT 436 11 "sigma(n)=  " }{XPPEDIT 18 0 "(p[1]^(a[1]+1)-1)/(
p[1]-1);" "6#*&,&)&%\"pG6#\"\"\",&&%\"aG6#\"\"\"\"\"\"\"\"\"F/F/\"\"\"
!\"\"F/,&&F'6#\"\"\"F/\"\"\"F2F2" }{TEXT 443 3 " * " }{XPPEDIT 18 0 "(
p[2]^(a[2]+1)-1)/(p[2]-1);" "6#*&,&)&%\"pG6#\"\"#,&&%\"aG6#\"\"#\"\"\"
\"\"\"F/F/\"\"\"!\"\"F/,&&F'6#\"\"#F/\"\"\"F2F2" }{TEXT 444 9 " * ...*
  " }{XPPEDIT 18 0 "(p[s]^(a[s]+1)-1)/(p[s]-1);" "6#*&,&)&%\"pG6#%\"sG
,&&%\"aG6#F)\"\"\"\"\"\"F.F.\"\"\"!\"\"F.,&&F'6#F)F.\"\"\"F1F1" }
{TEXT -1 10 "\nCalcular " }{TEXT 437 12 "sigma(19800)" }{TEXT 438 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "restart; with(numtheory):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "ifactor(19800);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 
"((2**(3+1)-1)/(2-1))*((3**(2+1)-1)/(3-1))*((5**(2+1)-1)/(5-1))*((11**
(1+1)-1)/(11-1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
422 13 "Ejercicio 2: " }{TEXT -1 39 "Determinar cu\341ntos n\372meros \+
de la forma:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 6 "2^p-1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 16 "son primos, con " }{TEXT 423 1 "p" }{TEXT -1 87 "
 tomando valores entre 1 y 30 (ambos inclusive). Determinar exactament
e los valores de " }{TEXT 424 1 "p" }{TEXT -1 22 " que los hacen primo
s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 112 "j:=0:\nfor i from 1 to 30 do \nif( isprime(2**i -1) \+
) then j:=j+1; \nprint(i) fi ;od; \nprint('numero-de-primos',j);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
425 13 "Ejercicio 3: " }{TEXT -1 83 "Determinar, si existen, todas las
 soluciones del sistema de congruencias siguiente:" }}{PARA 0 "" 0 "" 
{TEXT -1 11 "7x=3(mod11)" }}{PARA 0 "" 0 "" {TEXT -1 10 "5x=2(mod7)" }
}{PARA 0 "" 0 "" {TEXT -1 11 "3x=7(mod13)" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "msolve(7*x=3,11);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "msolve(5*x=2,7);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "msolve(3*x=7,13);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "chrem([2,6,11],[11,7,13]); \+
\n11*7*13;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 16 "La respuesta es " }{TEXT 430 10 "167 +1001k" }{TEXT -1 5 " con \+
" }{TEXT 431 1 "k" }{TEXT -1 18 " un n\372mero entero." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 426 13 "Ejer
cicio 4: " }{TEXT -1 35 "El m\351todo de cifrado conocido como " }
{TEXT 427 13 "transposici\363n" }{TEXT -1 103 " consiste en cambiar de
 lugar las letras de un mensaje para hacerlo incomprensible. Usando la
 librer\355a " }{TEXT 428 8 "combinat" }{TEXT -1 24 " descifrar el men
saje:  " }{TEXT 432 7 "aaivl. " }{TEXT -1 58 "(Sabiendo que se ha de r
ecibir una palabra en castellano.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart;\nwith(combinat):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "permute([a,a,i,v,l]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "La respuesta est\341 entre las
 posibles permutaciones del mensaje recibido, y solo tiene sentido la \+
palabra " }{TEXT 429 5 "avila" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Examen de Maple.
 Septiembre 2003." }}{PARA 265 "" 0 "" {TEXT -1 37 "Septiembre. Inform
\341tica de Sistemas. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 445 12 "Ejercicio 1:" }{TEXT -1 6 "  Sea " }
{TEXT 448 7 "G=(V,E)" }{TEXT -1 29 " un multigrafo no dirigido y " }
{TEXT 446 1 "M" }{TEXT -1 38 " una matriz de adyacencias asociada a " 
}{TEXT 449 1 "G" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(linalg):" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
7612 "M:=matrix([[1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0
, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1,
 1, 0, 1, 0, 1, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0
, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0,
 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1], [0, 0, 1, 0, 1, 0, 1, 0, 1, 1
, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
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, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0,
 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1], [1, 0
, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1,
 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, \+
1, 1], [1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1,
 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, \+
1, 1, 1, 1, 1, 0], [1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0,
 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, \+
0, 0, 1, 1, 1, 0, 1, 0, 1, 1], [1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1,
 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, \+
0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0], [0, 1, 1, 1, 0, 1, 1, 0, 0,
 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, \+
0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0,
 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, \+
1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1], [1,
 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, \+
1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0
, 1, 1], [0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, \+
1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0
, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, \+
0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0
, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, \+
0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0
, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1], [0, 0, 1, 0, 0, 1, 0, 0, \+
0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1
, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0], [1, 1, 1, 0, \+
1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1
, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1], [
1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1
, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1,
 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1
, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0,
 1, 0, 1, 1, 1, 1, 0], [1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1
, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0,
 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0], [1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0
, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1,
 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0], [0, 1, 0, 0, 1, 1, 1, 1
, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1,
 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1], [1, 0, 1, 0
, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0,
 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0],
 [1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0,
 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, \+
1, 0, 1, 1], [0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1,
 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, \+
0, 1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1,
 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, \+
1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1], [1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0,
 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, \+
0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1], [0, 1, 0, 0, 1, 0, 0,
 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, \+
0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0], [0, 1, 0,
 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, \+
0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0
], [0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, \+
0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1
, 0, 1, 1, 1], [0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, \+
1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0
, 1, 1, 1, 1, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, \+
0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0
, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, 1, 1, \+
0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1
, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0], [0, 1, 0, 1, 1, 1, \+
1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1
, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0], [0, 0, \+
0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0
, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1,
 1], [0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0
, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0,
 1, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1
, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1,
 1, 0, 1, 0, 1, 0, 0, 1, 1], [1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0
, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0,
 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1], [0, 1, 0, 0, 0, 1, 1, 0, 1, 0
, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1,
 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 0, 0
, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1,
 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1], [0, 0
, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0,
 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, \+
0, 1], [1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1,
 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, \+
1, 0, 1, 0, 1, 0], [1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0,
 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, \+
0, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0,
 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, \+
1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0], [1, 0, 1, 0, 1, 0, 1, 0, 1,
 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, \+
1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 1, 1,
 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, \+
1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1], [1,
 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, \+
1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1
, 0, 1], [0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, \+
0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0
, 0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, \+
0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0
, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, \+
1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1
, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1], [0, 1, 1, 1, 1, 0, 1, 0, \+
1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0
, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1]]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 450 77 "Determinar el n\372mero de v\351rtices \+
y aristas del grafo. Determinar si el grafo " }{TEXT 454 2 "G'" }
{TEXT 455 1 " " }{TEXT 456 10 "=(V',E'), " }{TEXT 457 6 "donde " }
{TEXT 458 4 "V'=V" }{TEXT 459 3 " y " }{TEXT 460 2 "E'" }{TEXT 461 49 
" es el resultado de quitar los lazos al conjunto " }{TEXT 462 3 "E, \+
" }{TEXT 463 18 "es un grafo simple" }{TEXT 478 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "El n\372mero de v\351rt
ices no es m\341s que el n\372mero de filas (o columnas) de la matriz \+
" }{TEXT 479 1 "M" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 453 1 " " }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rowdim(M);\ncoldim(M);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Los posib
les lazos del grafo " }{TEXT 480 1 "G" }{TEXT -1 63 " est\341n represe
ntados por unos en la diagonal. Como la diagonal:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "seq(M[i,i],i
=1..50);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
40 "tiene 4 unos, entonces hay cuatro lazos." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Para ver el n\372mero de arista
s, hay que contabilizar los lazos dos veces, entonces:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "d:=0:\n
for i from 1 to rowdim(M) do\nfor j from 1 to rowdim(M) do \nd:=d+M[i,
j] od:\nod:(d+4)/2;" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 65 "O, equivalentemente, s\363lo contabiliza
r las entradas de la matriz " }{TEXT 481 7 "M[i,j] " }{TEXT -1 10 "tal
es que " }{TEXT 482 1 "i" }{TEXT -1 57 " es menor o igual (o equivalen
temente mayor o igual) que " }{TEXT 483 2 "j." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "d:=0:\nfor i
 from 1 to rowdim(M) do\nfor j from i to rowdim(M) do \nd:=d+M[i,j] od
:\nod: d;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 107 "Para determinar si el grafo sin lazos es simple, hay que compr
obar que todas sus entradas son unos o ceros:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "S:=0:\nfor \+
i from 1 to rowdim(M) do\nfor j from 1 to rowdim(M) do \nif M[i,j]>1 t
hen S:=1: fi od:\nod: \nS;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 22 "Como as\355 es en efecto." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
447 12 "Ejercicio 2." }{TEXT -1 46 " Sea el sistema de congruencias de
finido por: " }{TEXT 464 2 "x " }{TEXT -1 27 "congruente con 5 m\363du
lo 7, " }{TEXT 465 1 "x" }{TEXT -1 30 " congruente con 3 m\363dulo 11 \+
y " }{TEXT 466 2 "x " }{TEXT -1 58 "congruente con 4 m\363dulo 17. Det
erminar c\372antas soluciones " }{TEXT 467 2 "x " }{TEXT -1 22 "del si
stema verifican " }{TEXT 468 15 "12345<x<196780." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Basta recorrer esos valor
es de la variable " }{TEXT 484 1 "x" }{TEXT -1 54 " e ir comprobando s
i en efecto verifican la condici\363n." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "c:=0:\nfor x from 12346
 to 196779 do\nif x mod 7=5 and x mod 11=3 and x mod 17=4 then \nc:=c+
1: fi: od: c; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 451 12 "Ejercicio 3." }{TEXT -1 
59 " Sea un experimento de Bernoulli con probabilidad de \351xito " }
{TEXT 452 6 "p=0.1 " }{TEXT 469 14 "que se repite " }{TEXT 471 1 "n" }
{TEXT 472 8 " veces (" }{TEXT 473 3 "n>2" }{TEXT 474 1 ")" }{TEXT 470 
2 ". " }{TEXT -1 149 "Determinar si la probabilidad de obtener exactam
ente dos \351xitos crece al aumentar el n\372mero de repeticiones del \+
experimento, esto es, al incrementar " }{TEXT 475 3 "n. " }{TEXT -1 
54 "Idem con la probabilidad de obtener al menos un \351xito." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "En un exp
erimento de Bernoulli que se repite " }{TEXT 485 1 "n" }{TEXT -1 76 " \+
veces tenemos que las probabilidades que se nos piden son, respectivam
ente " }{TEXT 486 2 "f " }{TEXT -1 2 "y " }{TEXT 487 3 "g, " }{TEXT 
-1 41 "entendidas como funciones de la variable " }{TEXT 488 1 "n" }
{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "with(combinat):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "f:=n->binomial(n,2)*(0.1)^2*(1-0.1)^(n-2);\ng:=n->1-0
.9^n;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 
"Observamos que: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "f(3);f(100);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 11 "Por lo que " }{TEXT 489 2 "f " }
{TEXT -1 28 "no es una funci\363n creciente." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Mientras que la derivada de la \+
funci\363n " }{TEXT 490 1 "g" }{TEXT -1 4 " es:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(g(n),n)
;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "por
 tanto positiva, as\355, en efecto la funci\363n " }{TEXT 491 2 "g " }
{TEXT -1 6 "crece." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 476 12 "Ejercicio 4." }{TEXT -1 26 " Verificar si en el grafo " 
}{TEXT 477 1 "G" }{TEXT -1 77 " del ejercicio 1 todo par de v\351rtice
s est\341 unido por un camino de longitud 2." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Basta tomar el cuadrado de la m
atriz " }{TEXT 492 1 "M" }{TEXT -1 0 "" }{TEXT 493 0 "" }{TEXT -1 48 "
 y verificar si todas sus entradas son no nulas." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "T:=evalm(M^2
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "c:=0:\nfor i from 1 \+
to rowdim(T) do\nfor j from 1 to coldim(T) do\nif T[i,j]=0 then c:=c+1
: \nfi:\nod: od:\nc;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Como as\355 ocurre en efecto." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "21" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }