Difference between revisions of "2000 USAMO Problems/Problem 3"

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=== Problem 3 ===
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=== Problem ===
  
 
A game of solitaire is played with <math>R</math> red cards, <math>W</math> white cards, and <math>B</math> 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 <math>R, W,</math> and <math>B,</math> the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
 
A game of solitaire is played with <math>R</math> red cards, <math>W</math> white cards, and <math>B</math> 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 <math>R, W,</math> and <math>B,</math> the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.

Revision as of 14:40, 22 April 2012

Problem

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.