1985 IMO 第 3 题

(NET 3) ${ }^{\mathrm{IMO} 3}$ The weight $w(p)$ of a polynomial $p, p(x)=\sum_{i=0}^{n} a_{i} x^{i}$, 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 $0 \leq i_{1}<i_{2}<\cdots<i_{n}$, the inequality $$w\left(q_{i_{1}}+\cdots+q_{i_{n}}\right) \geq w\left(q_{i_{1}}\right)$$ holds.

(NET 3) ${ }^{\mathrm{IMO} 3}$ 具有整数系数 $a_{i}$ 的多项式 $p, p(x)=\sum_{i=0}^{n} a_{i} x^{i}$ 的权重 $w(p)$ 定义为其奇数系数的数量。对于$i=0,1,2,\ldots$，令$q_{i}(x)=(1+x)^{i}$。证明对于任何有限序列 $0 \leq i_{1}<i_{2}<\cdots<i_{n}$，不等式 $$w\left(q_{i_{1}}+\cdots+q_{i_{n}}\right) \geq w\left(q_{i_{1}}\right)$$ 成立。