2006 APMO 第 5 题

In a circus, there are $n$ clowns who dress and paint themselves up using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set

of colours and no more than 20 clowns may use any one particular colour. Find the largest number $n$ of clowns so as to make the ringmaster's order possible.

在马戏团里，有 $n$ 小丑，他们用 12 种不同的颜色来打扮自己。每个小丑必须使用至少五种不同的颜色。有一天，马戏团的领班命令任何两个小丑都不能拥有完全相同的一套

颜色的数量不超过 20 个小丑可以使用任何一种特定颜色。找到最大数量的$n$小丑，以使马戏团指挥的命令成为可能。