试题(2020年9月国际数学奥林匹克

There are 4n pebbles of weights 1,2,3,…,4n. Each pebble is coloured in one of n colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:

● The total weights of both piles are the same.

● Each pile contains two pebbles of each colour.

有 4n 枚石子,重量分别为 1 , 2 , 3 , … , 4n .每一枚小石子都染了n种颜色之一,使得每种颜色的小石子恰有四枚.证明:可以把这些小石子分成两堆,且满足以下两个条件:

● 两堆小石子的总重量相同;

● 每堆中每种颜色的小石子各有两枚.

(匈牙利供题)

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关键词

证明;数学;定义;石子;平面几何;条件;表示;定理;存在;图;