For any polynomial with integer coefficients, the number of odd coefficients is denoted by . For let . Prove that if are integers satisfying , then:
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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: $$o(Q_{i_1}+Q_{i_2}+\ldots+Q_{i_n})\ge o(Q_{i_1}).$$
Source: Međunarodna matematička olimpijada, shortlist 1985
Školjka 
with integer coefficients, the number of odd coefficients is denoted by 
. For 
let 
. Prove that if 
are integers satisfying 
, then: 