5577: Mysterious Light
时间限制: 2 Sec 内存限制: 256 MB提交: 248 解决: 77
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题目描述
Snuke is conducting an optical experiment using mirrors and his new invention, the rifle of Mysterious Light.
Three mirrors of length N are set so that they form an equilateral triangle. Let the vertices of the triangle be a,b and c.
Inside the triangle, the rifle is placed at the point p on segment ab such that ap=X. (The size of the rifle is negligible.) Now, the rifle is about to fire a ray of Mysterious Light in the direction of bc.
The ray of Mysterious Light will travel in a straight line, and will be reflected by mirrors, in the same ways as "ordinary" light. There is one major difference, though: it will be also reflected by its own trajectory as if it is a mirror! When the ray comes back to the rifle, the ray will be absorbed.
The following image shows the ray's trajectory where N=5 and X=2.
It can be shown that the ray eventually comes back to the rifle and is absorbed, regardless of the values of N and X. Find the total length of the ray's trajectory.
Constraints
2≦N≦1012
1≦X≦N−1
N and X are integers.
Partial Points
300 points will be awarded for passing the test set satisfying N≦1000.
Another 200 points will be awarded for passing the test set without additional constraints.
Three mirrors of length N are set so that they form an equilateral triangle. Let the vertices of the triangle be a,b and c.
Inside the triangle, the rifle is placed at the point p on segment ab such that ap=X. (The size of the rifle is negligible.) Now, the rifle is about to fire a ray of Mysterious Light in the direction of bc.
The ray of Mysterious Light will travel in a straight line, and will be reflected by mirrors, in the same ways as "ordinary" light. There is one major difference, though: it will be also reflected by its own trajectory as if it is a mirror! When the ray comes back to the rifle, the ray will be absorbed.
The following image shows the ray's trajectory where N=5 and X=2.
Constraints
2≦N≦1012
1≦X≦N−1
N and X are integers.
Partial Points
300 points will be awarded for passing the test set satisfying N≦1000.
Another 200 points will be awarded for passing the test set without additional constraints.
输入
The input is given from Standard Input in the following format:N X
输出
Print the total length of the ray's trajectory.
样例输入
5 2
样例输出
12
提示
Refer to the image in the Problem Statement section. The total length of the trajectory is 2+3+2+2+1+1+1=12.
来源
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<iostream>
using namespace std;
int main()
{
long long n,x;
scanf("%lld%lld",&n,&x);
if(x>n/2) x=n-x;///x表示折射长度
long long tmp=n-x;///tmp表示折射区间
long long zhong;
long long ans=0;
while(x!=0)
{
ans+=(tmp/x)*x*3;///tmp/x表示有几个三角形
zhong=tmp%x;///表示下一组生成的三角形的长度
tmp=x;
x=zhong;
}
printf("%lld\n",ans);
return 0;
}