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Putnam 2012 A5

Source:

December 3, 2012

Putnamvectorlinear algebramatrixalgebrafunctiondomain

Problem Statement

Let

\mathbb{F}_p

denote the field of integers modulo a prime

p,

and let

n

be a positive integer. Let

v

be a fixed vector in

\mathbb{F}_p^n,

let

M

be an

n\times n

matrix with entries in

\mathbb{F}_p,

and define

G:\mathbb{F}_p^n\to \mathbb{F}_p^n

by

G^{(1)}(x)=G(x)

Let

G^{(k)}

denote the

k

-fold composition of

G

with itself, that is,

G^{(1)}(x)=G(x)

and

G^{(k+1)}(x)=G(G^{(k)}(x)).

Determine all pairs

p,n

for which there exist

v

and

M

such that the

p^n

vectors

G^{(k)}(0),

k=1,2,\dots,p^n

are distinct.

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