MathDB

Polynomial inequality - IMO Long List 1986

Source:

August 29, 2010

polynomialbinomial coefficientscombinatorial inequalitypascal s trianglecombinatoricsIMO ShortlistIMO 1985

Problem Statement

For any polynomial

P(x)=a_0+a_1x+\ldots+a_kx^k

with integer coefficients, the number of odd coefficients is denoted by

o(P)

. For

i-0,1,2,\ldots

let

Q_i(x)=(1+x)^i

. Prove that if

i_1,i_2,\ldots,i_n

are integers satisfying

0\le i_1<i_2<\ldots<i_n

, then:

o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}).