Question

Alpha 和 Beta 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。 Alpha 第一天尝试了 $300$ 分，得了 $160$ 分；第二天尝试了 $200$ 分，得了 $140$ 分。 Beta 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；Beta 每天的成功率（得分除以尝试分数）都低于 Alpha 当天的成功率。Alpha 两天总成功率为 $300/500 = 3/5$ 。 Beta 可能达到的最大两天总成功率为 $m/n$ ，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 $300/500 = 3/5.$ The largest possible two-day success ratio that Beta could have achieved is $m/n,$ 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

解答：

Alpha 两天的成功率分别为 $\frac{160}{300} = \frac{8}{15}$ 和 $\frac{140}{200} = \frac{7}{10}$ 。因为 $\frac{8}{15} \lt \frac{7}{10}$ ，Beta 每天的得分都小于当天尝试分数的 $\frac{7}{10}$ ，所以 Beta 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：Beta 第一天可在尝试 $2$ 分中得 $1$ 分（且 $\frac{1}{2} \lt \frac{8}{15}$ ），第二天在尝试 $498$ 分中得 $348$ 分（且 $\frac{348}{498} \lt \frac{7}{10}$ ，因为 $3480 \lt 3486$ ）。

因此 Beta 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

解答：

其他年份的第 5 题