Score : $600$ points

问题描述

有 $N$ 种类型的物品，每种物品都有无限多个复制品。第 $i$ 种物品的重量为 $A_i$，体积为 $B_i$，价值为 $C_i$。

等级为 $X$ 的高桥最多能携带总重量不超过 $X$、总体积也不超过 $X$ 的物品。在此条件下，他可以携带任意数量的同种物品，也可以选择不携带某些种类的物品。

令 $f(X)$ 表示等级为 $X$ 的高桥能够携带的最大物品总价值。可以证明极限 $\displaystyle\lim_{X\to \infty} \frac{f(X)}{X}$ 存在。求这个极限值。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ kinds of items, each with infinitely many copies. The $i$-th kind of item has a weight of $A_i$, a volume of $B_i$, and a value of $C_i$.

Level $X$ Takahashi can carry items whose total weight is at most $X$ and whose total volume is at most $X$. Under this condition, it is allowed to carry any number of items of the same kind, or omit some kinds of items.

Let $f(X)$ be the maximum total value of items Level $X$ Takahashi can carry. It can be proved that the limit $\displaystyle\lim_{X\to \infty} \frac{f(X)}{X}$ exists. Find this limit.

Constraints

• $1\leq N\leq 2\times 10^5$
• $10^8\leq A_i,B_i,C_i \leq 10^9$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $B_1$ $C_1$

$A_2$ $B_2$ $C_2$

$\vdots$

$A_N$ $B_N$ $C_N$

Output

Print the answer. Your output will be considered correct if the absolute or relative error from the judge's answer is at most $10^{-6}$.

Sample Input 1

2
100000000 200000000 100000000
200000000 100000000 100000000

Sample Output 1

0.6666666666666667

When $X=300000000$, Takahashi can carry items whose total weight is at most $300000000$ and whose total volume is at most $300000000$.

He can carry, for instance, one copy of the $1$-st item and one copy of the $2$-nd item. Then, the total value of the items is $100000000+100000000=200000000$. This is the maximum achievable value, so $\dfrac{f(300000000)}{300000000}=\dfrac{2}{3}$.

It can also be proved that $\displaystyle\lim_{X\to \infty} \frac{f(X)}{X}$ equals $\dfrac{2}{3}$. Thus, the answer is $0.6666666...$.

Sample Input 2

1
500000000 300000000 123456789

Sample Output 2

0.2469135780000000

update @ 2024/3/10 11:35:50