Question

一名网球运动员用获胜场数除以总比赛场数来计算胜率。某个周末开始时，她的胜率恰为 $0.500$ 。周末期间，她参加了四场比赛，三胜一负。周末结束时，她的胜率大于 $0.503$ 。在这个周末开始前，她最多可能赢过多少场比赛？

A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly $0.500.$ During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than $0.503.$ What’s the largest number of matches she could’ve won before the weekend began?

答案

答案：164

知识点：比与比例不等式百分数

难度评级：1640

小提示：

若她最初赢了 $w$ 场，胜率为 $0.500$ 意味着她共参加了 $2w$ 场比赛

If she had $w$ wins initially, a $0.500$ ratio means she had played $2w$ matches

大提示：

先把最终胜率写成严格不等式，再求最大的整数 $w$

Translate the final ratio into a strict inequality before taking the largest integer $w$

解答：

若她最初赢了 $w$ 场，那么她共参加了 $2w$ 场比赛。最终条件为 $\frac{w+3}{2w+4}\gt\frac{503}{1000}\text{。}$ 交叉相乘得 $1000w+3000\gt1006w+2012$ ，所以 $6w\lt988$ ，从而 $w\lt164\frac23$ 。满足条件的最大整数为 $164$ 。

If she initially had $w$ wins, then she had played $2w$ matches. The final condition is $\frac{w+3}{2w+4}\gt\frac{503}{1000}.$ Cross-multiplication gives $1000w+3000\gt1006w+2012,$ so $6w\lt988$ and $w\lt164\frac23.$ The largest possible integer is $164.$

1992 AIME 第 3 题

答案：164

难度评级：1640

小提示：

大提示：

解答：

其他年份的第 3 题