2000 USAMO 第 3 题

A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.

纸牌游戏使用 $R$ 红牌、$W$ 白牌和 $B$ 蓝牌。玩家一次打出一张所有牌。每场比赛他都会累积一个罚球。如果他打出一张蓝牌，那么他将被收取罚金，罚金的数额是他手中仍然有白牌的数量。如果他出一张白牌，那么他将被处以手上剩余红牌数量两倍的处罚。如果他出示红牌，那么他将被处以手上剩余蓝牌数量三倍的处罚。找出作为 $R、W、$ 和 $B、$ 的函数，玩家可以累积的最小总惩罚以及实现该最小值的所有方法。