Question

去年 Isabella 参加了 $7$ 次数学测验，得到 $7$ 个互不相同的分数，每个分数都是 $91$ 到 $100$ 之间（含端点）的整数。每次测验后，她都注意到到目前为止所有测验分数的平均数是整数。她第七次测验的分数是 $95$ 。她第六次测验的分数是多少？

Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100,$ inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95.$ What was her score on the sixth test?

$92$

$94$

$96$

$98$

$100$

答案：E

知识点：整除性平均数极限情形界定

难度评级：2040

解答：

令 $S$ 为七个分数之和。则 $S$ 是 $7$ 的倍数且 $658 \le S \le 679$ 所以 $S \in \{658, 665, 672, 679\}$ 因为前六次测验后的平均数是整数， $S - 95$ 必须是 $6$ 的倍数，这迫使 $S = 665$ 。因此前六个分数之和为 $570$ ，它是 $5$ 的倍数。前五次测验后的平均数也是整数，所以前五个分数之和是 $5$ 的倍数，于是第六个分数也是 $5$ 的倍数。由于所有分数互不相同且第七个分数为 $95$ ，第六个分数必须是 $100$ 。

所以正确答案是 E。

Let $S$ be the sum of all seven scores. Then $S$ is a multiple of $7$ with $658 \le S \le 679,$ so $S \in \{658, 665, 672, 679\}.$ Since the average after six tests is an integer, $S - 95$ is a multiple of $6,$ which forces $S = 665.$ Then the first six scores sum to $570,$ a multiple of $5;$ the average after five tests is an integer, so the first five scores also sum to a multiple of $5,$ making the sixth score a multiple of $5.$ Since all scores differ and the seventh is $95,$ the sixth must be $100.$

Thus, the correct answer is E.

2017 AMC 12B 第 21 题

答案：E

难度评级：2040

解答：

其他年份的第 21 题