Ayoub's function

1000ms

262144K

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?

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".

Sample Input:

Sample Output:

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.

Informação

Provedor Codeforces

Código CF1301C