问题 B:
时间限制: 2 Sec 内存限制: 256 MB
提交: 209 解决: 55
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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.
#include<iostream>
#define LL long long
using namespace std;
int main()
{
LL n,x;
while(cin>>n>>x)
{
LL t=x;
LL sum=n;
x=min(t,n-t);//分析可知x于n-x答案相同(相当于旋转了)
LL y=max(t,n-t);//这里只计算x小的那种情况
while(x>=1)
{
if(y%x==0)//如果y能整除x刚好能射出
{
sum+=(y/x)*2*x;
sum-=x;
break;
}
else{ //否则的话更新反射后的x,y的值。
sum+=(y/x)*2*x;
LL te=x;
x=y%x;
y=te;
}
}
cout<<sum<<endl;
}
}