xtu 1268 Strange Optimization 湘潭邀请赛I

Strange Optimization

Accepted : 67 Submit : 289Time Limit : 1000 MS Memory Limit : 65536 KB

 

Strange Optimization

Bobo is facing a strange optimization problem. Given n,m , he is going to find a real number α such thatf(12+α) is maximized, where f(t)=mini,j∈Z|in−jm+t| . Help him!

Note: It can be proved that the result is always rational.

Input

The input contains zero or more test cases and is terminated by end-of-file.

Each test case contains two integers n,m .

• 1≤n,m≤109
• The number of tests cases does not exceed 10⁴ .

Output

For each case, output a fraction p/q which denotes the result.

Sample Input

1 11 2

Sample Output

1/21/4

Note

For the first sample, α=0 maximizes the function

题目大意

题目给你一个函数，让你求每一段值域最小值中的最大值

思路

首先根据我们数学的直觉，会先把i/n-j/m通分，得出

 得出（i*m-j*n）/n*m   然后显然可以提出gcd（n，m）因子

就相当于是k*gcd（n，m）/n*m，k为整数，就是个等差数列，然后就相当于去找符合条件的首项，那么由于是绝对值，，不用考虑最小边界的问题，那么容易得出首项为二分之一公差答案，然后把答案化到最简就行了，

代码：

#include<cstdio>#include<algorithm>#include<cstring>#include<map>#include<vector>#include<iostream>#include<sstream>#define LL long longusing namespace std;LL gcd(LL a,LL b){    return b==0?a:gcd(b,a%b);}int main(){    LL n,m;    while(~scanf("%lld%lld",&n,&m))    {        LL d = gcd(n,m);        LL d2  = gcd(d,2*n*m);        cout<<1<<'/'<<2*m*n/d2<<endl;    }}