#P1301C. Ayoub's function
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ID: 5202
Type: RemoteJudge
1000ms
256MiB
Tried: 0
Accepted: 0
Difficulty: (None)
Uploaded By:
Hydro
Tags> Ayoub's function
No submission language available for this problem.
Description
Ayoub thinks that he is a very smart person, so he created a function $f(s)$, where $s$ is a binary string (a string which contains only symbols "0" and "1"). The function $f(s)$ is equal to the number of substrings in the string $s$ that contains at least one symbol, that is equal to "1".
More formally, $f(s)$ is equal to the number of pairs of integers $(l, r)$, such that $1 \leq l \leq r \leq |s|$ (where $|s|$ is equal to the length of string $s$), such that at least one of the symbols $s_l, s_{l+1}, \ldots, s_r$ is equal to "1".
For example, if $s = $"01010" then $f(s) = 12$, because there are $12$ such pairs $(l, r)$: $(1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 4), (4, 5)$.
Ayoub also thinks that he is smarter than Mahmoud so he gave him two integers $n$ and $m$ and asked him this problem. For all binary strings $s$ of length $n$ which contains exactly $m$ symbols equal to "1", find the maximum value of $f(s)$.
Mahmoud couldn't solve the problem so he asked you for help. Can you help him?
The input consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 10^5$) — the number of test cases. The description of the test cases follows.
The only line for each test case contains two integers $n$, $m$ ($1 \leq n \leq 10^{9}$, $0 \leq m \leq n$) — the length of the string and the number of symbols equal to "1" in it.
For every test case print one integer number — the maximum value of $f(s)$ over all strings $s$ of length $n$, which has exactly $m$ symbols, equal to "1".
Input
The input consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 10^5$) — the number of test cases. The description of the test cases follows.
The only line for each test case contains two integers $n$, $m$ ($1 \leq n \leq 10^{9}$, $0 \leq m \leq n$) — the length of the string and the number of symbols equal to "1" in it.
Output
For every test case print one integer number — the maximum value of $f(s)$ over all strings $s$ of length $n$, which has exactly $m$ symbols, equal to "1".
Samples
5
3 1
3 2
3 3
4 0
5 2
4
5
6
0
12
Note
In the first test case, there exists only $3$ strings of length $3$, which has exactly $1$ symbol, equal to "1". These strings are: $s_1 = $"100", $s_2 = $"010", $s_3 = $"001". The values of $f$ for them are: $f(s_1) = 3, f(s_2) = 4, f(s_3) = 3$, so the maximum value is $4$ and the answer is $4$.
In the second test case, the string $s$ with the maximum value is "101".
In the third test case, the string $s$ with the maximum value is "111".
In the fourth test case, the only string $s$ of length $4$, which has exactly $0$ symbols, equal to "1" is "0000" and the value of $f$ for that string is $0$, so the answer is $0$.
In the fifth test case, the string $s$ with the maximum value is "01010" and it is described as an example in the problem statement.