2024 APMO 第 4 题

Prove that for every positive integer $t$ there is a unique permutation $a_{0}, a_{1}, \ldots, a_{t-1}$ of $0,1, \ldots, t-$ 1 such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2 a_{i}}$ is odd and $2 a_{i} \neq t+i$.

证明对于每个正整数 $t$，存在 $0,1, \ldots, t-$ 1 的唯一排列 $a_{0}, a_{1}, \ldots, a_{t-1}$，使得对于每个 $0 \leq i \leq t-1$，二项式系数 $\binom{t+i}{2 a_{i}}$ 为奇数，$2 a_{i} \neq t+i$。