URAL 2025. Line Fighting （math）

2025. Line Fighting

Time limit: 1.0 second

Memory limit: 64 MB

Boxing, karate, sambo… The audience is sick of classic combat sports. That is why a popular sports channel launches a new competition format based on the traditional Russian entertainment called line fighting.There can be from
2 to k teams taking part in a competition, and there are n fighters altogether in all the teams. Before the competition starts, the fighters are divided into teams: each fighter becomes a member of exactly one team.Two fighters fight each
other if they are members of different teams. The organizers believe that the more the number of fights between fighters, the higher the popularity of a competition will be. Help the organizers to distribute fighters between teams so as to maximize the number
of fights and output this number.

Input

The first line contains the number of tests T (1 ≤
T ≤ 10). In each of the following T lines you are given a test:integers
n and k separated with a space (2 ≤ k ≤ n ≤ 10⁴).

Output

For each test output the answer (one integer) in a separate line.

Sample

input	output
3 6 3 5 5 4 2	12 10 4

Problem Author: Alexey Danilyuk

Problem Source: Ural Regional School Programming Contest 2014

解析：组合数学。因为组内不能打比赛，这就相当于在全部人都能比赛的基础上去掉了各个组间能打的比赛次数。

首先，比赛次数最多的情况肯定是尽可能地将人数均分，这种比赛数是最多的。

AC代码：

#include <bits/stdc++.h>
using namespace std;

int main(){
    #ifdef sxk
        freopen("in.txt", "r", stdin);
    #endif // sxk

    int T, n, k, ans;
    scanf("%d", &T);
    while(T --){
        scanf("%d%d", &n, &k);
        ans = n * (n - 1) / 2;     //全部人两两之间打比赛的次数
        if(n != k){
            int foo = n / k;
            int cnt = n % k;       //均分后剩余cnt个人，再均分。则会出现cnt个人数多1的组
            ans -= cnt * ((foo + 1) * foo / 2);   //去掉人数较多的cnt组的总次数
            ans -= (k - cnt) * (foo * (foo - 1) / 2);   //去掉人数较少的总次数
        }
        printf("%d\n", ans);
    }
    return 0;
}