//****************************************************************************************************

//

//  求和为n的连续正整数序列 - C++ - by Chimomo

//

//  题目： 输入一个正整数n，输出全部和为n的连续正整数序列。比如：输入15，因为1+2+3+4+5=4+5+6=7+8=15，所以输出3个连续序列1-5、4-6和7-8。

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//  Answer: Suppose n = i+(i+1)+...+(j-1)+j, then n = (i+j)(j-i+1)/2 = (j*j-i*i+i+j)/2 => j^2+j+(i-i^2-2n) = 0 => j = (sqrt(1-4(i-i^2-2n))-1)/2 => j = (sqrt(4i^2+8n-4i+1)-1)/2.

//          We know 1 <= i < j <= n/2+1, so for each i in [1,n/2], do this arithmetic to check if there is a integer answer.

//

//  Note: 二次函数 ax^2+bx+c=0 的求根公式为： x = (-b±sqrt(b^2-4ac)) / 2a。

//

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#include <iostream>

#include <cassert>

#include <stack>

#include <math.h>

using namespace std ;

int FindConsecutiveSequence(int n)

{

	int count = 0;

	for (int i = 1; i <= n/2; i++)

	{

		double sqroot = sqrt(4*i*i + 8*n - 4*i + 1);

		int floor = sqroot;

		if(sqroot == floor)

		{

			cout << i << "-" << (sqroot - 1) / 2 << endl;

			count++;

		}

	}

	return count;

}

int main()

{

	int count = FindConsecutiveSequence(15);

	cout << "Totally " << count << " sequences found." << endl;

	return 0;

}

// Output:

/*

1-5

4-6

7-8

Totally 3 sequences found.

*/