2010 USAMO 第 6 题

A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number $N$ of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.

黑板包含 68 对非零整数。假设对于每个正整数 $k$，至多一对 $(k, k)$ 和 $(-k, -k)$ 写在黑板上。一名学生擦除了 136 个整数中的一些，但条件是任何两个被擦除的整数相加不得为 0。然后，该学生为 68 对中至少有一个整数被擦除的每对得分 1 分。通过证明，确定学生可以保证得分的最大分数 $N$，无论黑板上写的是哪 68 对。