2023 AIME 答案与解析 - 第一期

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原题目

Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is where and are relatively prime positive integers. Find

绿树教育中心独家解析

先选取 5 对座位（1男1女），随后任意安置，故有

则 。

原题目

Positive real numbers and satisfy the equations

The value of is where and are relatively prime positive integers. Find

绿树教育中心独家解析

令 . 因此，问题中给出的方程组可以改写为

则， and . 则,

则，所求为 .

原题目

A plane contains lines, no of which are parallel. Suppose that there are points where exactly lines intersect, points where exactly lines intersect, points where exactly lines intersect, points where exactly lines intersect, and no points where more than lines intersect. Find the number of points where exactly lines intersect.

绿树教育中心独家解析

所有交点： 个，对于 条线的交点，多算 次。故

原题目

The sum of all positive integers such that is a perfect square can be written as where and are positive integers. Find

绿树教育中心独家解析

我们有

要使分数成为平方，它需要每个素数都是偶数幂。则 必须包含 。另外， 可以包含 的任意偶数次幂。则

故答案为 .

原题目

Let be a point on the circle circumscribing square that satisfies and Find the area of

绿树教育中心独家解析

原题目

Alice knows that red cards and black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is where and are relatively prime positive integers. Find

绿树教育中心独家解析

最佳策略：选择余下个数最多的种类，相同时取红。分类讨论得

原题目

Call a positive integer extra-distinct if the remainders when is divided by and are distinct. Find the number of extra-distinct positive integers less than .

绿树教育中心独家解析

在 时枚举，有 。故总数为

原题目

Rhombus has There is a point on the incircle of the rhombus such that the distances from to the lines and are and respectively. Find the perimeter of

绿树教育中心独家解析

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