For any polynomial
![P(x)=a_0+a_1x+\ldots+a_kx^k](/media/m/1/f/e/1fecb0e46016011f5b13970d86bf3748.png)
with integer coefficients, the number of odd coefficients is denoted by
![o(P)](/media/m/0/1/f/01f1fa6347d74d1b63f2ab09c43d8313.png)
. For
![i-0,1,2,\ldots](/media/m/2/b/c/2bcb3bca5e00ecd68b8e5d0a18bbe168.png)
let
![Q_i(x)=(1+x)^i](/media/m/f/6/7/f679008ef25eb12e9ddecd0765a2f0ea.png)
. Prove that if
![i_1,i_2,\ldots,i_n](/media/m/3/2/3/323de5d9e9d54a43ca3ca147cfd09729.png)
are integers satisfying
![0\le i_1<i_2<\ldots<i_n](/media/m/a/e/e/aeea9f9dc2ca41e44e3f423ea7ea92da.png)
, then:
%V0
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: $$o(Q_{i_1}+Q_{i_2}+\ldots+Q_{i_n})\ge o(Q_{i_1}).$$