A16445 | Split Plus K
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题目描述
[eliteLAQ - Desert Ruins](https://soundcloud.com/lux-gg-198448038/desert-ruins)
⠀
There are $n$ positive integers $a_1, a_2, \dots, a_n$ on a blackboard. You are also given a positive integer $k$ . You can perform the following operation some (possibly $0$ ) times:
- choose a number $x$ on the blackboard;
- erase one occurrence of $x$ ;
- write two positive integers $y$ , $z$ such that $y+z = x+k$ on the blackboard.
Is it possible to make all the numbers on the blackboard equal? If yes, what is the minimum number of operations you need?
⠀
There are $n$ positive integers $a_1, a_2, \dots, a_n$ on a blackboard. You are also given a positive integer $k$ . You can perform the following operation some (possibly $0$ ) times:
- choose a number $x$ on the blackboard;
- erase one occurrence of $x$ ;
- write two positive integers $y$ , $z$ such that $y+z = x+k$ on the blackboard.
Is it possible to make all the numbers on the blackboard equal? If yes, what is the minimum number of operations you need?
输入格式
Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 10^4$ ). The description of the test cases follows.
The first line of each test case contains two integers $n$ , $k$ ( $1 \le n \le 2 \cdot 10^5$ , $1 \leq k \leq 10^{12}$ ) — the number of integers initially on the blackboard and the constant $k$ .
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_i \le 10^{12}$ ) — the initial state of the blackboard.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .
The first line of each test case contains two integers $n$ , $k$ ( $1 \le n \le 2 \cdot 10^5$ , $1 \leq k \leq 10^{12}$ ) — the number of integers initially on the blackboard and the constant $k$ .
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_i \le 10^{12}$ ) — the initial state of the blackboard.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .
输出格式
For each test case, output a single line containing an integer: the minimum number of operations you need to make all the numbers on the blackboard equal, or $-1$ if it is impossible.
输入输出样例
输入 #1
9 2 1 3 4 2 3 7 11 3 10 100 40 100 2 1 1 2 2 2 1 2 1 327869541 327869541 5 26250314066 439986238782 581370817372 409476934981 287439719777 737637983182 5 616753575719 321037808624 222034505841 214063039282 441536506916 464097941819 5 431813672576 393004301966 405902283416 900951084746 672201172466 518769038906
输出 #1
3 1 4 -1 -1 0 3119 28999960732 -1
In the first test case, $k = 1$ . You can make all the numbers on the blackboard equal to $2$ with the following operations:
- Erase $x = 4$ and write $(y, z) = (2, 3)$ . Note that $y+z=x+k$ . The blackboard now contains the multiset $\{3, 2, 3\}$ .
- Erase $x = 3$ and write $(y, z) = (2, 2)$ . Note that $y+z=x+k$ . The blackboard now contains $\{2, 2, 2, 3\}$ .
- Erase $x = 3$ and write $(y, z) = (2, 2)$ . Note that $y+z=x+k$ . The blackboard now contains $\{2, 2, 2, 2, 2\}$ .
This makes all the numbers equal in $3$ operations. It can be shown that you cannot make all the numbers equal in less than $3$ operations.
In the second test case, $k = 3$ . You can make all the numbers on the blackboard equal to $7$ with the following operation:
- Erase $x = 11$ and write $(y, z) = (7, 7)$ . Note that $y+z=x+k$ . The blackboard now contains $\{7, 7, 7\}$ .
In the third test case, $k = 10$ . You can make all the numbers on the blackboard equal to $40$ with the following operations:
- Erase $x = 100$ and write $(y, z) = (70, 40)$ . Note that $y+z=x+k$ . The blackboard now contains $\{70, 40, 40, 100\}$ .
- Erase $x = 70$ and write $(y, z) = (40, 40)$ . Note that $y+z=x+k$ . The blackboard now contains $\{40, 40, 40, 40, 100\}$ .
- Erase $x = 100$ and write $(y, z) = (40, 70)$ . Note that $y+z=x+k$ . The blackboard now contains $\{40, 40, 40, 40, 40, 70\}$ .
- Erase $x = 70$ and write $(y, z) = (40, 40)$ . Note that $y+z=x+k$ . The blackboard now contains $\{40, 40, 40, 40, 40, 40, 40\}$ .
In the fourth and in the fifth test case, you can show that it is impossible to make all the numbers on the blackboard equal.
- Erase $x = 4$ and write $(y, z) = (2, 3)$ . Note that $y+z=x+k$ . The blackboard now contains the multiset $\{3, 2, 3\}$ .
- Erase $x = 3$ and write $(y, z) = (2, 2)$ . Note that $y+z=x+k$ . The blackboard now contains $\{2, 2, 2, 3\}$ .
- Erase $x = 3$ and write $(y, z) = (2, 2)$ . Note that $y+z=x+k$ . The blackboard now contains $\{2, 2, 2, 2, 2\}$ .
This makes all the numbers equal in $3$ operations. It can be shown that you cannot make all the numbers equal in less than $3$ operations.
In the second test case, $k = 3$ . You can make all the numbers on the blackboard equal to $7$ with the following operation:
- Erase $x = 11$ and write $(y, z) = (7, 7)$ . Note that $y+z=x+k$ . The blackboard now contains $\{7, 7, 7\}$ .
In the third test case, $k = 10$ . You can make all the numbers on the blackboard equal to $40$ with the following operations:
- Erase $x = 100$ and write $(y, z) = (70, 40)$ . Note that $y+z=x+k$ . The blackboard now contains $\{70, 40, 40, 100\}$ .
- Erase $x = 70$ and write $(y, z) = (40, 40)$ . Note that $y+z=x+k$ . The blackboard now contains $\{40, 40, 40, 40, 100\}$ .
- Erase $x = 100$ and write $(y, z) = (40, 70)$ . Note that $y+z=x+k$ . The blackboard now contains $\{40, 40, 40, 40, 40, 70\}$ .
- Erase $x = 70$ and write $(y, z) = (40, 40)$ . Note that $y+z=x+k$ . The blackboard now contains $\{40, 40, 40, 40, 40, 40, 40\}$ .
In the fourth and in the fifth test case, you can show that it is impossible to make all the numbers on the blackboard equal.
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评测结果：Accepted