【CODEFORCES】 B. Friends and Presents

最新推荐文章于 2024-07-17 22:13:38 发布

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最新推荐文章于 2024-07-17 22:13:38 发布

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分类专栏： Codeforces 数学文章标签： Codeforces 数学

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解决如何在限制条件下为两位朋友分配互不相同的整数礼物的问题。利用二分查找与容斥原理，确保分配的礼物数符合特定条件且不超过指定范围。

You have two friends. You want to present each of them several positive integers. You want to present cnt1 numbers to the first friend and cnt2 numbers to the second friend. Moreover, you want all presented numbers to be distinct, that also means that no number should be presented to both friends.

In addition, the first friend does not like the numbers that are divisible without remainder by prime number x. The second one does not like the numbers that are divisible without remainder by prime number y. Of course, you're not going to present your friends numbers they don't like.

Your task is to find such minimum number v, that you can form presents using numbers from a set 1, 2, ..., v. Of course you may choose not to present some numbers at all.

A positive integer number greater than 1 is called prime if it has no positive divisors other than 1 and itself.

Input

The only line contains four positive integers cnt1, cnt2, x, y (1 ≤ cnt1, cnt2 < 109; cnt1 + cnt2 ≤ 109; 2 ≤ x < y ≤ 3·104) — the numbers that are described in the statement. It is guaranteed that numbers x, y are prime.

Output

Print a single integer — the answer to the problem.

Sample test(s)

input
3 1 2 3

output
5

input
1 3 2 3

output
4

Note

In the first sample you give the set of numbers {1, 3, 5} to the first friend and the set of numbers {2} to the second friend. Note that if you give set {1, 3, 5} to the first friend, then we cannot give any of the numbers 1, 3, 5 to the second friend.

In the second sample you give the set of numbers {3} to the first friend, and the set of numbers {1, 2, 4} to the second friend. Thus, the answer to the problem is 4.

题解：这一题是2分加容斥原理，二分整个区间，这里取的上界是2e9。这样一个人不收x的倍数，另一个人不收y的倍数，对于一个解n来说，n-n/x就是第一个人最多能接受的个数。第二个人同理。又因为所有的数中可能有些数即是x的倍数又是y的倍数，所以要用到容斥原理，算出n中总共有多少数是两个人都能接受的。算出这些之后与cnt比较就可以二分了。

#include <iostream>
#include <cstdio>
#include <cstring>

using namespace std;

long long cnt1,cnt2,x,y,t;

long long gcd(long long x,long long y)
{
	if (y==0) return x;
	else return gcd(y,x%y);
}

int main()
{
    scanf("%I64d%I64d%I64d%I64d",&cnt1,&cnt2,&x,&y);
    long long l=1,r=2000000000,k=x*y/gcd(x,y);
    while (l<r)
    {
        if (l==r)
        {
            printf("%I64d",r);
            return 0;
        }
        t=(l+r)/2;
        if (cnt1<=t-t/x && cnt2<=t-t/y && t>=cnt1+cnt2+t/k ) r=t;
        else l=t+1;
    }
    printf("%I64d",r);
    return 0;
}