A16042 | The Good Array
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题目描述
You are given two integers $n$ and $k$ .
An array $a_1, a_2, \ldots, a_n$ of length $n$ , consisting of zeroes and ones is good if for all integers $i$ from $1$ to $n$ both of the following conditions are satisfied:
- at least $\lceil \frac{i}{k} \rceil$ of the first $i$ elements of $a$ are equal to $1$ ,
- at least $\lceil \frac{i}{k} \rceil$ of the last $i$ elements of $a$ are equal to $1$ .
Here, $\lceil \frac{i}{k} \rceil$ denotes the result of division of $i$ by $k$ , rounded up. For example, $\lceil \frac{6}{3} \rceil = 2$ , $\lceil \frac{11}{5} \rceil = \lceil 2.2 \rceil = 3$ and $\lceil \frac{7}{4} \rceil = \lceil 1.75 \rceil = 2$ .
Find the minimum possible number of ones in a good array.
An array $a_1, a_2, \ldots, a_n$ of length $n$ , consisting of zeroes and ones is good if for all integers $i$ from $1$ to $n$ both of the following conditions are satisfied:
- at least $\lceil \frac{i}{k} \rceil$ of the first $i$ elements of $a$ are equal to $1$ ,
- at least $\lceil \frac{i}{k} \rceil$ of the last $i$ elements of $a$ are equal to $1$ .
Here, $\lceil \frac{i}{k} \rceil$ denotes the result of division of $i$ by $k$ , rounded up. For example, $\lceil \frac{6}{3} \rceil = 2$ , $\lceil \frac{11}{5} \rceil = \lceil 2.2 \rceil = 3$ and $\lceil \frac{7}{4} \rceil = \lceil 1.75 \rceil = 2$ .
Find the minimum possible number of ones in a good array.
输入格式
Each test contains multiple test cases. The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.
The only line of each test case contains two integers $n$ , $k$ ( $2 \le n \le 100$ , $1 \le k \le n$ ) — the length of array and parameter $k$ from the statement.
The only line of each test case contains two integers $n$ , $k$ ( $2 \le n \le 100$ , $1 \le k \le n$ ) — the length of array and parameter $k$ from the statement.
输出格式
For each test case output one integer — the minimum possible number of ones in a good array.
It can be shown that under the given constraints at least one good array always exists.
It can be shown that under the given constraints at least one good array always exists.
输入输出样例
输入 #1
7 3 2 5 2 9 3 7 1 10 4 9 5 8 8
输出 #1
2 3 4 7 4 3 2
In the first test case, $n = 3$ and $k = 2$ :
- Array $[ \, 1, 0, 1 \, ]$ is good and the number of ones in it is $2$ .
- Arrays $[ \, 0, 0, 0 \, ]$ , $[ \, 0, 1, 0 \, ]$ and $[ \, 0, 0, 1 \, ]$ are not good since for $i=1$ the first condition from the statement is not satisfied.
- Array $[ \, 1, 0, 0 \, ]$ is not good since for $i=1$ the second condition from the statement is not satisfied.
- All other arrays of length $3$ contain at least $2$ ones.
Thus, the answer is $2$ .
In the second test case, $n = 5$ and $k = 2$ :
- Array $[ \, 1, 1, 0, 0, 1 \, ]$ is not good since for $i=3$ the second condition is not satisfied.
- Array $[ \, 1, 0, 1, 0, 1 \, ]$ is good and the number of ones in it is $3$ .
- It can be shown that there is no good array with less than $3$ ones, so the answer is $3$ .
In the third test case, $n = 9$ and $k = 3$ :
- Array $[ \, 1, 0, 1, 0, 0, 0, 1, 0, 1 \, ]$ is good and the number of ones in it is $4$ .
- It can be shown that there is no good array with less than $4$ ones, so the answer is $4$ .
In the fourth test case, $n = 7$ and $k = 1$ . The only good array is $[ \, 1, 1, 1, 1, 1, 1, 1\, ]$ , so the answer is $7$ .
- Array $[ \, 1, 0, 1 \, ]$ is good and the number of ones in it is $2$ .
- Arrays $[ \, 0, 0, 0 \, ]$ , $[ \, 0, 1, 0 \, ]$ and $[ \, 0, 0, 1 \, ]$ are not good since for $i=1$ the first condition from the statement is not satisfied.
- Array $[ \, 1, 0, 0 \, ]$ is not good since for $i=1$ the second condition from the statement is not satisfied.
- All other arrays of length $3$ contain at least $2$ ones.
Thus, the answer is $2$ .
In the second test case, $n = 5$ and $k = 2$ :
- Array $[ \, 1, 1, 0, 0, 1 \, ]$ is not good since for $i=3$ the second condition is not satisfied.
- Array $[ \, 1, 0, 1, 0, 1 \, ]$ is good and the number of ones in it is $3$ .
- It can be shown that there is no good array with less than $3$ ones, so the answer is $3$ .
In the third test case, $n = 9$ and $k = 3$ :
- Array $[ \, 1, 0, 1, 0, 0, 0, 1, 0, 1 \, ]$ is good and the number of ones in it is $4$ .
- It can be shown that there is no good array with less than $4$ ones, so the answer is $4$ .
In the fourth test case, $n = 7$ and $k = 1$ . The only good array is $[ \, 1, 1, 1, 1, 1, 1, 1\, ]$ , so the answer is $7$ .
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评测结果：Accepted