Question 3
The weight $w(p)$ of a polynomial $p$, $p(x)=\sum_{i=0}^n a_ix^i$, with integer coefficients $a_i$ is defined as the number of its odd coefficients. For $i=0,1,2,\ldots$, let $q_i(x)=(1+x)^i$. Prove that for any finite sequence $0\le i_1\lt i_2\lt \cdots\lt i_n$ the inequality
$$w(q_{i_1}+\cdots+q_{i_n})\ge w(q_{i_1})$$
holds.
$$w(q_{i_1}+\cdots+q_{i_n})\ge w(q_{i_1})$$
holds.