2000 CMO 第 6 题

A test contains $5$ multiple choice questions which have $4$ options in each. Suppose each examinee chose one option for each question. There exists a number $n$ , such that for any $n$ sheets among $2000$ sheets of answer papers, there are $4$ sheets of answer papers such that any two of them have at most $3$ questions with the same answers. Find the minimum value of $n$ .

测试包含 5 美元的多项选择题，每个问题有 4 美元的选项。假设每位考生为每个问题选择一个选项。存在一个数字 $n$ ，使得对于 $2000$ 张答卷中的任何 $n$ 张，都有 $4$ 张答卷，使得其中任何两个最多有 $3$ 问题具有相同的答案。找到 $n$ 的最小值。