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2023/02/09阅读：70主题：全栈蓝

# 2023 AIME 答案与解析 - 第一期

### 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

### 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

. 因此，问题中给出的方程组可以改写为

### 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.

### 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

### 原题目

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

### 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

### 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 .

### 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

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