CF1654C.Alice and the Cake

普及/提高-

通过率：0%

AC君温馨提醒

该题目为【codeforces】题库的题目，您提交的代码将被提交至codeforces进行远程评测，并由ACGO抓取测评结果后进行展示。由于远程测评的测评机由其他平台提供，我们无法保证该服务的稳定性，若提交后无反应，请等待一段时间后再进行重试。

题目描述

Alice has a cake, and she is going to cut it. She will perform the following operation $n-1$ times: choose a piece of the cake (initially, the cake is all one piece) with weight $w\ge 2$ and cut it into two smaller pieces of weight $\lfloor\frac{w}{2}\rfloor$ and $\lceil\frac{w}{2}\rceil$ ( $\lfloor x \rfloor$ and $\lceil x \rceil$ denote floor and ceiling functions, respectively).

After cutting the cake in $n$ pieces, she will line up these $n$ pieces on a table in an arbitrary order. Let $a_i$ be the weight of the $i$ -th piece in the line.

You are given the array $a$ . Determine whether there exists an initial weight and sequence of operations which results in $a$ .

输入格式

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

The first line of each test case contains a single integer $n$ ( $1 \le n \le 2 \cdot 10^5$ ).

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_i \le 10^9$ ).

It is guaranteed that the sum of $n$ for all test cases does not exceed $2 \cdot 10^5$ .

输出格式

For each test case, print a single line: print YES if the array $a$ could have resulted from Alice's operations, otherwise print NO.

You may print each letter in any case (for example, YES, Yes, yes, yEs will all be recognized as positive answer).

输入输出样例

输入#1

14
1
327
2
869 541
2
985214736 985214737
3
2 3 1
3
2 3 3
6
1 1 1 1 1 1
6
100 100 100 100 100 100
8
100 100 100 100 100 100 100 100
8
2 16 1 8 64 1 4 32
10
1 2 4 7 1 1 1 1 7 2
10
7 1 1 1 3 1 3 3 2 3
10
1 4 4 1 1 1 3 3 3 1
10
2 3 2 2 1 2 2 2 2 2
4
999999999 999999999 999999999 999999999

输出#1

YES
NO
YES
YES
NO
YES
NO
YES
YES
YES
YES
NO
NO
YES

说明/提示

In the first test case, it's possible to get the array $a$ by performing $0$ operations on a cake with weight $327$ .

In the second test case, it's not possible to get the array $a$ .

In the third test case, it's possible to get the array $a$ by performing $1$ operation on a cake with weight $1\,970\,429\,473$ :

Cut it in half, so that the weights are $[985\,214\,736, 985\,214\,737]$ .

Note that the starting weight can be greater than $10^9$ .In the fourth test case, it's possible to get the array $a$ by performing $2$ operations on a cake with weight $6$ :

Cut it in half, so that the weights are $[3,3]$ .
Cut one of the two pieces with weight $3$ , so that the new weights are $[1, 2, 3]$ which is equivalent to $[2, 3, 1]$ up to reordering.

首页