【codeforces】Non-square Equation（数学推导）

一时失言乱红尘 2022-07-15 07:57 184阅读 0赞

time limit per test

1 second

memory limit per test

256 megabytes

input

standard input

output

standard output

Let's consider equation:

*x*2 + *s*(*x*)·*x* - *n* = 0,

where *x*, *n* are positive integers, *s*(*x*) is the function, equal to the sum of digits of number *x* in the decimal number system.

You are given an integer *n*, find the smallest positive integer root of equation *x*, or else determine that there are no such roots.

Input

A single line contains integer *n* (1 ≤ *n* ≤ 1018) — the equation parameter.

Please, do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use cin, cout streams or the%I64d specifier.

Output

Print -1, if the equation doesn't have integer positive roots. Otherwise print such smallest integer *x* (*x* > 0), that the equation given in the statement holds.

Sample test(s)

input

output

input

110

output

input

output

-1

Note

In the first test case *x* = 1 is the minimum root. As *s*(1) = 1 and 12 + 1·1 - 2 = 0.

In the second test case *x* = 10 is the minimum root. As *s*(10) = 1 + 0 = 1 and 102 + 1·10 - 110 = 0.

In the third test case the equation has no roots.

本题用二分是不行的，因为y=x+s(x)\*x并不是单调递增的。

我们看直接暴力肯定会超时的。所以要看怎么优化，这就需要用到一些数学知识了。看下面这组公式：

![Center][]

我们估算x不会超过10^9,所以s(x)不会超过81；

由上述公式我们就可以对s(x)的值遍历一遍，求出相应的x,再检验此x值是否符合所给公式。从而避免了直接对x遍历超时的情况

#include<cstdio>
    #include<iostream>
    #include<cstring>
    #include<cmath>
    #include<algorithm>
    #define MAXN 10000000000
    using namespace std;
    typedef long long LL;
    LL solve(LL x){
    	LL t=0;
    	while(x){
    		t=t+x%10;
    		x=x/10;
    	}
    	return t;
    }
    int main(){
    	LL n;
    	while(scanf("%lld",&n)!=EOF){
    		LL ans=MAXN;
    		for(int i=1;i<=81;i++){
    			LL x=sqrt(i*i/4+n)-i/2;
    			LL s=solve(x);
    			if(x*x+x*s-n==0){
    				ans=min(ans,x);
    				break;
    			}
    		}
    		if(ans!=MAXN)
    			printf("%lld\n",ans);
    		else
    			printf("-1\n");
    	}
    	return 0;
    }

[Center]: /images/20220715/025e5b49addf46318d5d7171727e17a6.png