Score : $400$ points

问题描述

给定一个包含正整数的序列 $A = (A_1, A_2, \dots, A_N)$，其长度为 $N$，以及两个整数 $x$ 和 $y$。
判断是否可以在 $xy$ 坐标平面上放置 $N+1$ 个点 $p_1, p_2, \dots, p_N, p_{N+1}$，满足以下所有条件。允许将两个或多个点放在相同的坐标位置。

• $p_1 = (0, 0)$。
• $p_2 = (A_1, 0)$。
• $p_{N+1} = (x, y)$。
• 点 $p_i$ 和 $p_{i+1}$ 之间的距离为 $A_i$。（$1 \leq i \leq N$）
• 线段 $p_i p_{i+1}$ 和 $p_{i+1} p_{i+2}$ 形成一个 $90$ 度角。（$1 \leq i \leq N - 1$）

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a sequence $A = (A_1, A_2, \dots, A_N)$ of length $N$ consisting of positive integers, and integers $x$ and $y$.
Determine whether it is possible to place $N+1$ points $p_1, p_2, \dots, p_N, p_{N+1}$ in the $xy$-coordinate plane to satisfy all of the following conditions. (It is allowed to place two or more points at the same coordinates.)

• $p_1 = (0, 0)$.
• $p_2 = (A_1, 0)$.
• $p_{N+1} = (x, y)$.
• The distance between the points $p_i$ and $p_{i+1}$ is $A_i$. ($1 \leq i \leq N$)
• The segments $p_i p_{i+1}$ and $p_{i+1} p_{i+2}$ form a $90$ degree angle. ($1 \leq i \leq N - 1$)

Constraints

• $2 \leq N \leq 10^3$
• $1 \leq A_i \leq 10$
• $|x|, |y| \leq 10^4$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $x$ $y$

$A_1$ $A_2$ $\dots$ $A_N$

Output

If it is possible to place $p_1, p_2, \dots, p_N, p_{N+1}$ to satisfy all of the conditions in the Problem Statement, print Yes; otherwise, print No.

Sample Input 1

3 -1 1
2 1 3

Sample Output 1

Yes

The figure below shows a placement where $p_1 = (0, 0)$, $p_2 = (2, 0)$, $p_3 = (2, 1)$, and $p_4 = (-1, 1)$. All conditions in the Problem Statement are satisfied.

Sample Input 2

5 2 0
2 2 2 2 2

Sample Output 2

Yes

Letting $p_1 = (0, 0)$, $p_2 = (2, 0)$, $p_3 = (2, 2)$, $p_4 = (0, 2)$, $p_5 = (0, 0)$, and $p_6 = (2, 0)$ satisfies all the conditions. Note that multiple points may be placed at the same coordinates.

Sample Input 3

4 5 5
1 2 3 4

Sample Output 3

No

Sample Input 4

3 2 7
2 7 4

Sample Output 4

No

Sample Input 5

10 8 -7
6 10 4 1 5 9 8 6 5 1

Sample Output 5

Yes

update @ 2024/3/10 11:32:49