2020 IMO Shortlist C6

$4 n$ coins of weights $1,2,3, \ldots, 4 n$ are given. Each coin is colored in one of $n$ colors and there are four coins of each color. Show that all these coins can be partitioned into two sets with the same total weight, such that each set contains two coins of each color. (Hungary)

给出了 $4 n$ 硬币，重量为 $1,2,3, \ldots, 4 n$。每枚硬币都涂有 $n$ 种颜色中的一种，每种颜色有四枚硬币。证明所有这些硬币可以分成总重量相同的两组，使得每组包含每种颜色的两枚硬币。 (匈牙利)