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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.
In a 2 × 3 table there are 6 1 × 1 squares and 2 2 × 2 squares. That is equal to 8 squares in total.
题意：给你x，问你多少种n*m的情况使得，在当前这个矩形内小正方形的个数为x
题解：
列式推： sigma(k=1,k=min(n,m))(n-k+1)*(m-k+1)=x；
我们枚举n，得到m
///1085422276 #include<bits/stdc++.h> using namespace std; typedef long long ll; #define mem(a) memset(a,0,sizeof(a)) #define pb push_back inline ll read() { ll x=0,f=1;char ch=getchar(); while(ch<‘0‘||ch>‘9‘){ if(ch==‘-‘)f=-1;ch=getchar(); } while(ch>=‘0‘&&ch<=‘9‘){ x=x*10+ch-‘0‘;ch=getchar(); }return x*f; } //**************************************** const int N=100000+350; #define maxn 100000+5 ll x; int main() { x=read(); vector<pair<ll, ll> > ans; ll sum=0; for(ll i=1; sum<=x; i++) { sum+=1ll*i*i; long long d=x-sum; long long k=1LL*i*(i+1)/2; if(d%k==0) { ans.push_back({i, d/k+i}); ans.push_back({d/k+i, i}); } } sort(ans.begin(), ans.end()); ans.resize(unique(ans.begin(), ans.end())-ans.begin()); printf("%d\n", ans.size()); for(ll i=0;i<ans.size();i++) printf("%I64d %I64d\n", ans[i].first, ans[i].second); return 0; }
Codeforces Round #332 (Div. 2)D. Spongebob and Squares 数学
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原文地址：http://www.cnblogs.com/zxhl/p/4984252.html
