Codeforces 599D:Spongebob and Squares
Spongebob is already tired trying to reason his weird actions and calculations, so he simply asked you to find all pairs of n and m, such that there are exactly x distinct squares in the table consisting of n rows and m columns. For example, in a 3 × 5 table there are 15squares with side one, 8 squares with side two and 3 squares with side three. The total number of distinct squares in a 3 × 5 table is15 + 8 + 3 = 26.
The first line of the input contains a single integer x (1 ≤ x ≤ 1018) — the number of squares inside the tables Spongebob is interested in.
First print a single integer k — the number of tables with exactly x distinct squares inside.
Then print k pairs of integers describing the tables. Print the pairs in the order of increasing n, and in case of equality — in the order of increasing m.
26
6 1 26 2 9 3 5 5 3 9 2 26 1
2
2 1 2 2 1
8
4 1 8 2 3 3 2 8 1
In a 1 × 2 table there are 2 1 × 1 squares. So, 2 distinct squares in total.
![](http://codeforces.com/predownloaded/fe/87/fe87d2f734cb907de7bec68808baba9556dc9ad9.png)
In a 2 × 3 table there are 6 1 × 1 squares and 2 2 × 2 squares. That is equal to 8 squares in total.
![](http://codeforces.com/predownloaded/d0/ba/d0ba7b2da6d50c0980c091718c70ab619aa53776.png)
题意是给定一个X,问那些矩形中含有的正方形总数等于X。
这题当时没时间做了,(太弱。。。)后面补的。
官方题解:
第一点:n*m里面的正方形数量就是sum((n-i)*(m-i)),i从1到n-1啊。。。在纸上画几次就明白了。
第二点:从1到n的平方和等于n(n+1)(2n+1)/6。。。
然后就是枚举n,求m。
代码:
#pragma warning(disable:4996) #include <iostream> #include <algorithm> #include <cmath> #include <vector> #include <string> #include <cstring> #include <map> using namespace std; typedef long long ll; const int maxn = 2000005; ll x; ll a[maxn]; ll b[maxn]; int main() { //freopen("i.txt", "r", stdin); //freopen("o.txt", "w", stdout); int flag; ll i, len, num, n, m, temp; cin >> x; flag = -1; num = 0; len = 2 * pow((double)x, ((double)1 / (double)3)); for (i = 1; i <= len+1; i++) { temp = 6 * x + i*i*i - i; n = i*i + i; if ((temp % (3 * n) == 0) && (i <= temp / (3 * n))) { a[num] = i; b[num] = temp / (3 * n); if (a[num] == b[num]) { flag = num; } num++; } } if (flag == -1) { cout << num * 2 << endl; for (i = 0; i < num; i++) { cout << a[i] << " " << b[i] << endl; } for (i = num-1; i >= 0; i--) { cout << b[i] << " " << a[i] << endl; } } else { cout << num * 2 - 1 << endl; for (i = 0; i < num; i++) { cout << a[i] << " " << b[i] << endl; } for (i = num - 1; i >= 0; i--) { if (flag == i) continue; cout << b[i] << " " << a[i] << endl; } } //system("pause"); return 0; }