Question

阿尔法和贝塔都参加了一个为期两天的解题竞赛。第二天结束时，每人尝试的问题总分都是 $500$ 分。阿尔法第一天得到 $160$ 分，尝试的问题总分为 $300$ 分；第二天得到 $140$ 分，尝试的问题总分为 $200$ 分。贝塔第一天尝试的总分不是 $300$ 分，并且两天每天都得到正整数分数；贝塔每天的成功率（得分除以尝试分数）都低于阿尔法当天的成功率。阿尔法两天总成功率为 $\frac{300}{500} = \frac{3}{5}$ 。贝塔可能达到的最大两天总成功率为 $\frac{m}{n}$ ，其中 $m$ 和 $n$ 是互质正整数。求 $m + n$ ？

Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of $500$ points. Alpha scored $160$ points out of $300$ points attempted on the first day, and scored $140$ points out of $200$ points attempted on the second day. Beta, who did not attempt $300$ points on the first day, had a positive integer score on each of the two days, and Beta’s daily success ratio (points scored divided by points attempted) on each day was less than Alpha’s on that day. Alpha’s two-day success ratio was $\frac{300}{500} = \frac{3}{5}.$ The largest possible two-day success ratio that Beta could have achieved is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$

答案

答案：849

知识点：比与比例不等式极端原理

难度评级：2480

小提示：

贝塔两天每天的成功率都低于 $\frac{7}{10}$ ，所以贝塔的总得分小于 $\frac{7}{10}$ 乘以 $500$

Both of Beta’s daily ratios are below $\frac{7}{10},$ so Beta’s total score is less than $\frac{7}{10}$ of $500$

大提示：

说明略低于 $350$ 的上界可以达到：第一天得到 $1$ 分，而尝试的问题总分为 $2$ 分

Show the bound just below $350$ is attainable: try $1$ point out of $2$ attempted on day one

解答：

阿尔法两天的成功率分别为 $\frac{160}{300} = \frac{8}{15}$ 和 $\frac{140}{200} = \frac{7}{10}$ 。因为 $\frac{8}{15} \lt \frac{7}{10}$ ，贝塔每天的得分都小于当天尝试分数的 $\frac{7}{10}$ ，所以贝塔的总得分小于 $\frac{7}{10} \cdot 500 = 350$ ，因而至多为 $349$ 。

总分 $349$ 可以达到：贝塔第一天可得 $1$ 分，尝试的问题总分为 $2$ 分（且 $\frac{1}{2} \lt \frac{8}{15}$ ）；第二天可得 $348$ 分，尝试的问题总分为 $498$ 分（且 $\frac{348}{498} \lt \frac{7}{10}$ ，因为 $3480 \lt 3486$ ）。

因此贝塔可能的最大两天总成功率是 $\frac{349}{500}$ ，由于 $349$ 是质数，这已经是最简分数，所以 $m + n = 349 + 500 = 849$ 。

Alpha’s daily ratios were $\frac{160}{300} = \frac{8}{15}$ and $\frac{140}{200} = \frac{7}{10}.$ Since $\frac{8}{15} \lt \frac{7}{10},$ Beta’s score was less than $\frac{7}{10}$ of the points attempted on each day, so Beta’s total score was less than $\frac{7}{10} \cdot 500 = 350,$ hence at most $349.$

A total of $349$ is achievable: Beta can score $1$ out of $2$ points attempted on day one (and $\frac{1}{2} \lt \frac{8}{15}$ ) and $348$ out of $498$ on day two (and $\frac{348}{498} \lt \frac{7}{10}$ because $3480 \lt 3486$ ).

So Beta’s largest possible two-day ratio is $\frac{349}{500},$ which is in lowest terms since $349$ is prime, and $m + n = 349 + 500 = 849.$

2004 AIME I 第 5 题

答案：849

难度评级：2480

小提示：

大提示：

解答：

其他年份的第 5 题