CF刷题遇到的一些数学题

Many Perfect Squares

题面翻译

有 $n$ 个数 $(1 \le n \le 50)$ 。 它们分别是 $a_1,a_2,...,a_n$ 。你需要选择一个数 $x$ ，使得 $x$ 在 $0$ 至 $10^{18}$ 之内，并且 $a_1 + x,a_2 + x,a_3 + x ... a_n +x$ 中有尽可能多的完全平方数。询问当 $x$ 最优时,有多少个数是完全平方数。

题目描述

You are given a set $a_1, a_2, \ldots, a_n$ of distinct positive integers.

We define the squareness of an integer $x$ as the number of perfect squares among the numbers $a_1 + x, a_2 + x, \ldots, a_n + x$ .

Find the maximum squareness among all integers $x$ between $0$ and $10^{18}$ , inclusive.

Perfect squares are integers of the form $t^2$ , where $t$ is a non-negative integer. The smallest perfect squares are $0, 1, 4, 9, 16, \ldots$ .

输入格式

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 50$ ). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $1 \le n \le 50$ ) — the size of the set.

The second line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ in increasing order ( $1 \le a_1 < a_2 < \ldots < a_n \le 10^9$ ) — the set itself.

It is guaranteed that the sum of $n$ over all test cases does not exceed $50$ .

输出格式

For each test case, print a single integer — the largest possible number of perfect squares among $a_1 + x, a_2 + x, \ldots, a_n + x$ , for some $0 \le x \le 10^{18}$ .

样例输入 #1

4
5
1 2 3 4 5
5
1 6 13 22 97
1
100
5
2 5 10 17 26

样例输出 #1

2
5
1
2

In the first test case, for $x = 0$ the set contains two perfect squares: $1$ and $4$ . It is impossible to obtain more than two perfect squares.

In the second test case, for $x = 3$ the set looks like $4, 9, 16, 25, 100$ , that is, all its elements are perfect squares.

思路

注意到n其实很小，答案至少为1，因为可以使得任意一个数变成平方数

现在我们考虑加一个$x$，可以使得至少两个数变为平方数到情况:

$a[i]+x=p^2$

$a[j]+x=q^2$

两式相减得到：$a[i]-a[j]=p^2-q^2$

于是：$a[i]-a[j]=(p+q)(p-q)$,这里$(p+q)和(p-q)$显然是具有相同奇偶性的

因为$n$很小，所以我们可以考虑枚举$a[i]-a[j]$,即$n^2$,设$diff=a[i]-a[j]$,可以先对$a$数组进行排序，降序排列

然后去枚举$diff$的因子$t=(p-q)$，得到如下结果：

$p-q=t$

$p+q=\frac{diff}{t}$

解方程得到$p=\frac{(t+\frac{diff}{t})}{2}$，因此$x=p^2-a[i]$

得到$x$后，再去判断数组$a$中的每个数是否都为平方数。

代码

#include <bits/stdc++.h>

#define int long long
#define yes cout << "YES" << endl;
#define no cout << "NO" << endl;
#define IOS cin.tie(0), cout.tie(0), ios::sync_with_stdio(false);
#define cxk 0
#define debug(s, x) if (cxk) cout << "#debug:(" << s << ")=" << x << endl;
using namespace std;

bool check(int x) {
    int t = sqrt(x);
    return t * t == x;
}

void solve() {
    int n;
    cin >> n;
    vector<int> a(n + 1);
    for (int i = 1; i <= n; i++) cin >> a[i];
    sort(a.begin() + 1, a.end());
    reverse(a.begin() + 1, a.end());
    int res = 1;
    for (int i = 1; i <= n; i++) {
        for (int j = i + 1; j <= n; j++) {
            int diff = a[i] - a[j];
            for (int t = 1; t * t <= diff; t++) {
                if (diff % t != 0) continue;
                if (t % 2 != (diff / t) % 2) continue; //两个因子奇偶性不同
                int p = (diff / t + t) / 2;
                int x = p * p - a[i];
                if (x < 0) continue;
                int cnt = 0;
                for (int k = 1; k <= n; k++) {
                    debug("a[k]+x", a[k] + x);
                    if (check(a[k] + x)) cnt++;
                }
                res = max(res, cnt);
            }
        }
    }
    cout << res << endl;
}

signed main() {
    IOS
#ifndef ONLINE_JUDGE
    freopen("../test.in", "r", stdin);
    freopen("../test.out", "w", stdout);
#endif
    int _ = 1;
    cin >> _;
    while (_--) solve();
    return 0;
}