HMMT 二月 2012 · 代数 · 第 2 题

HMMT February 2012 — Algebra — Problem 2

You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?

解析

You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards? Answer: 168 If there are B blue cards, then each red card contributes 1 + 2 B points (one for itself and two for each blue card) and each yellow card contributes 3 B points. Thus, if B > 1, it is optimal to change all red cards to yellow cards. When B = 0, the maximum number of points is 15. When B = 1, the number of points is always 42. When B > 1, the number of points is 3 BY , where Y is the number of yellow cards. Since B + Y = 15, the desired maximum occurs when B = 7 and Y = 8, which gives 168 points.