【Luogu P5515】[MtOI2019]灵梦的计算器

链接:

洛谷

博客园

题目大意:

求出 (lfloor n^a+n^b floor) 相等时,(n) 的上下界的差值。

正文:

如果不取整,函数 (f(x)=x^a+x^b) 的图象是这样的:

我们要找的是最大的 (x_1-x_2)

现在我们知道 (f(x_1)-f(x_2) ightarrow 1),即它趋向于 (1)。我们知道斜率的公式是 (frac{y_1-y_2}{x_1-x_2}=frac{f(x_1)-f(x_2)}{x_1-x_2}),我们可以通过斜率 (k) 倒推 (x_1-x_2=frac{f(x_1)-f(x_2)}{k})

而求斜率 (k),可以求导。设 (u(x)=x^a,v(x)=x^b),则 (f(x)=u(x)+v(x))

[egin{aligned}k=f'(x)&=u'(x)+v'(x)\ &=(x^a)'+(x^b)'\ &=ax^{a-1}+bx^{b-1}end{aligned}]

然后答案 (x_1-x_2=frac{0.dot{9}}{k})

代码:

inline ll READ()
{
	ll x = 0, f = 1;
	char c = getchar();
	while (c != '-' && (c < '0' || c > '9')) c = getchar();
	if (c == '-') f = -f, c = getchar();
	while (c >= '0' && c <= '9') x = (x << 3) + (x << 1) + c - '0', c = getchar();
	return x * f;
}

namespace Mker
{
//  Powered By Kawashiro_Nitori
//  Made In Gensokyo, Nihon
	#define uint unsigned int
	uint sd;int op;
	inline void init() {scanf("%u %d", &sd, &op);}
	inline uint uint_rand()
	{
		sd ^= sd << 13;
		sd ^= sd >> 7;
		sd ^= sd << 11;
		return sd;
	}
	inline double get_n()
	{
		double x = (double) (uint_rand() % 100000) / 100000;
		return x + 4;
	}
	inline double get_k()
	{
		double x = (double) (uint_rand() % 100000) / 100000;
		return (x + 1) * 5;
	}
	inline void read(double &n,double &a, double &b)
	{
		n = get_n(); a = get_k();
		if (op) b = a;
		else b = get_k(); 
	}
}

int t; 
double n, a, b, k, ans;

int main()
{
	t = READ();
	Mker::init();
	for (; t--; )
	{
		Mker::read(n, a, b);
		k = a * pow(n, a - 1) + b * pow(n, b - 1);
		ans += 0.9999999 / k;
	}
	printf ("%.5lf", ans);
	return 0;
}
原文地址:https://www.cnblogs.com/GJY-JURUO/p/14773786.html