3

Part of 1997 Irish Math Olympiad

Problems(2)

Source: Ireland 1997

Find all polynomials

p(x)

satisfying the equation: (x\minus{}16)p(2x)\equal{}16(x\minus{}1)p(x) for all

x

algebrapolynomialalgebra unsolved

Source: Ireland 1997

A

\{ 0,1,2,...,1997 \}

containing more than

1000

elements. Prove that either

A

contains a power of

2

(that is, a number of the form

2^k

with k\equal{}0,1,2,...) or there exist two distinct elements

a,b \in A

such that a\plus{}b is a power of

2

pigeonhole principlecombinatorics unsolvedcombinatorics