Score : $500$ points

问题陈述

存在一个初始体力为 $N$ 的怪物。
当怪物的体力大于等于1时，Takahashi会重复攻击该怪物。

Takahashi的每次攻击有$\frac{P}{100}$的概率减少怪物体力2点，以及$1-\frac{P}{100}$的概率减少怪物体力1点。

求在怪物体力降至0或以下之前，攻击次数的期望值，结果对$998244353$取模（具体见注释）。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There is a monster with initial stamina $N$.
Takahashi repeatedly attacks the monster while the monster's stamina remains $1$ or greater.

An attack by Takahashi reduces the monster's stamina by $2$ with probability $\frac{P}{100}$ and by $1$ with probability $1-\frac{P}{100}$.

Find the expected value, modulo $998244353$ (see Notes), of the number of attacks before the monster's stamina becomes $0$ or less.

Notes

We can prove that the sought expected value is always a finite rational number. Moreover, under the Constraints of this problem, when the value is represented as $\frac{P}{Q}$ by two coprime integers $P$ and $Q$, we can show that there exists a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Print such $R$.

Constraints

• $1 \leq N \leq 2\times 10^5$
• $0 \leq P \leq 100$
• All values in the input are integers.

Input

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

$N$ $P$

Output

Find the expected value, modulo $998244353$, of the number of Takahashi's attacks.

Sample Input 1

3 10

Sample Output 1

229596204

An attack by Takahashi reduces the monster's stamina by $2$ with probability $\frac{10}{100}=\frac{1}{10}$ and by $1$ with probability $\frac{100-10}{100}=\frac{9}{10}$.

• The monster's initial stamina is $3$.
• After the first attack, the monster's stamina is $2$ with probability $\frac{9}{10}$ and $1$ with probability $\frac{1}{10}$.
• After the second attack, the monster's stamina is $1$ with probability $\frac{81}{100}$, $0$ with probability $\frac{18}{100}$, and $-1$ with probability $\frac{1}{100}$. With probability $\frac{18}{100}+\frac{1}{100}=\frac{19}{100}$, the stamina becomes $0$ or less, and Takahashi stops attacking after two attacks.
• If the stamina remains $1$ after two attacks, the monster's stamina always becomes $0$ or less by the third attack, so he stops attacking after three attacks.

Therefore, the expected value is $2\times \frac{19}{100}+3\times\left(1-\frac{19}{100}\right)=\frac{281}{100}$. Since $229596204 \times 100 \equiv 281\pmod{998244353}$, print $229596204$.

Sample Input 2

5 100

Sample Output 2

3

Takahashi's attack always reduces the monster's stamina by $2$. After the second attack, the stamina remains $5-2\times 2=1$, so the third one is required.

Sample Input 3

280 59

Sample Output 3

567484387

update @ 2024/3/10 11:46:21