1978 IMO 第 1 题

(BUL 1) The set $M=\{1,2, \ldots, 2 n\}$ is partitioned into $k$ nonintersecting subsets $M_{1}, M_{2}, \ldots, M_{k}$, where $n \geq k^{3}+k$. Prove that there exist even numbers $2 j_{1}, 2 j_{2}, \ldots, 2 j_{k+1}$ in $M$ that are in one and the same subset $M_{i}$ $(1 \leq i \leq k)$ such that the numbers $2 j_{1}-1,2 j_{2}-1, \ldots, 2 j_{k+1}-1$ are also in one and the same subset $M_{j}(1 \leq j \leq k)$.

(BUL 1) 集合 $M=\{1,2, \ldots, 2 n\}$ 被划分为 $k$ 个不相交的子集 $M_{1}, M_{2}, \ldots, M_{k}$，其中 $n \geq k^{3}+k$。证明 $M$ 中存在偶数 $2 j_{1}, 2 j_{2}, \ldots, 2 j_{k+1}$ 位于同一个子集 $M_{i}$ $(1 \leq i \leq k)$ 中，使得数字 $2 j_{1}-1,2 j_{2}-1, \ldots, 2 j_{k+1}-1$ 也在同一个子集中子集 $M_{j}(1 \leq j \leq k)$。