Question

一个杰出委员会的成员正在选主席，每位成员都给 $27$ 位候选人中的一位投一票。对每位候选人来说，其所得票数的准确百分比至少比其所得票数小 $1$ 。委员会成员人数的最小可能值是多少？

The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?

答案

答案：134

知识点：百分数不等式极端原理

难度评级：2920

解答：

设成员人数为 $t$ 得到 $n$ 票的候选人的得票百分比为 $\frac{100n}{t}$ ，因此条件为 $\frac{100n}{t} \le n - 1$ ，整理得 $n(t - 100) \ge t$ 。这迫使 $t \gt 100$ ，并且 $n \ge \frac{t}{t - 100}.$

若 $t \le 133$ ，则 $\frac{t}{t - 100} \ge \frac{133}{33} \gt 4$ ，所以每位候选人都至少需要 $5$ 票，总票数至少为 $27 \cdot 5 = 135 \gt t$ ，不可能。

当 $t = 134$ 时，每位候选人需要 $n \ge \frac{134}{34}$ ，即至少 $4$ 票，这是可以做到的：让 $26$ 位候选人各得 $4$ 票，另一个候选人得 $30$ 。票。确实 $\frac{400}{134} \approx 2.99 \le 3$ 且 $\frac{3000}{134} \approx 22.4 \le 29$ 。所以最小可能成员人数为 $134$ 。

Let $t$ be the number of members. A candidate with $n$ votes has percentage $\frac{100n}{t},$ so the condition is $\frac{100n}{t} \le n - 1,$ which rearranges to $n(t - 100) \ge t.$ This forces $t \gt 100$ and $n \ge \frac{t}{t - 100}.$

If $t \le 133,$ then $\frac{t}{t - 100} \ge \frac{133}{33} \gt 4,$ so every candidate needs at least $5$ votes, and the total is at least $27 \cdot 5 = 135 \gt t$ — impossible.

For $t = 134,$ each candidate needs $n \ge \frac{134}{34},$ i.e. at least $4$ votes, and this is achievable: let $26$ candidates receive $4$ votes each and one receive $30.$ Indeed $\frac{400}{134} \approx 2.99 \le 3$ and $\frac{3000}{134} \approx 22.4 \le 29.$ So the smallest possible number of members is $134.$

2003 AIME II 第 12 题

答案：134

难度评级：2920

解答：

其他年份的第 12 题