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#8swx02. Yes or No

时间限制:2 s       空间限制:250 MiB       标签: 数学 概率期望 组合数学 组合计数 AtCoder AGC 

算法难度等级:3       思维难度等级:9       实现难度等级:4

本题来源于:AtCoder Grand Contest 019 Task F


N+MN+M 个问题,其中有 NN 个问题的答案是 YESMM 个问题的答案是 NO。当你回答一个问题之后,会知道这个问题的答案,求最优策略下期望对多少。

答案对 998244353998244353 取模。


You are participating in a quiz with N+MN+M questions and Yes/No answers.

It's known in advance that there are NN questions with answer Yes and MM questions with answer No, but the questions are given to you in random order.

You have no idea about correct answers to any of the questions. You answer questions one by one, and for each question you answer, you get to know the correct answer immediately after answering.

Suppose you follow a strategy maximizing the expected number of correct answers you give.

Let this expected number be P/QP/Q , an irreducible fraction. Let M=998244353M=998244353 . It can be proven that a unique integer RR between 00 and M1M-1 exists such that P=Q×RP=Q\times R modulo MM , and it is equal to P×Q1P\times Q^{-1} modulo MM , where Q1Q^{-1} is the modular inverse of QQ . Find RR .


Input is given from Standard Input in the following format: NMN\quad M


Let P/QP/Q be the expected number of correct answers you give if you follow an optimal strategy, represented as an irreducible fraction. Print P×Q1P\times Q^{-1} modulo 998244353998244353 .


1 1

样例解释 1

There are two questions. You may answer randomly to the first question, and you'll succeed with 50% probability. Then, since you know the second answer is different from the first one, you'll succeed with 100% probability.

The expected number of your correct answers is 32\dfrac 32 . Thus, P=3P=3 , Q=2Q=2 , Q1=499122177Q^{-1}=499122177 (modulo 998244353998244353 ), and P×Q1=499122178P\times Q^{-1}=499122178 (again, modulo 998244353998244353 ).

2 2

样例解释 2

The expected number of your correct answers is 176\dfrac{17}{6} .

3 4

样例解释 3

The expected number of your correct answers is 16935\dfrac{169}{35} .

10 10
42 23


  • 1N, M500,0001\leq N,\ M\leq 500,000
  • Both NN and MM are integers.

Partial Score

  • 15001500 points will be awarded for passing the testset satisfying N=MN=M and 1N,M1051\leq N, M\leq 10^5 .