Problem:


Score : 2000 points

Problem Statement

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

It's known in advance that there are N questions with answer Yes and M 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/Q, an irreducible fraction. Let M = 998244353.
It can be proven that a unique integer R between 0 and M - 1 exists such that P = Q \times R modulo M,
and it is equal to P \times Q^{-1} modulo M, where Q^{-1} is the modular inverse of Q.
Find R.

Constraints

1 \leq N, M \leq 500,000

Both N and M are integers.

Partial Score

1500 points will be awarded for passing the testset satisfying N = M and 1 \leq N, M \leq 10^5.

Input

Input is given from Standard Input in the following format:

N M

Output

Let P/Q be the expected number of correct answers you give if you follow an optimal strategy, represented as an irreducible fraction.
Print P \times Q^{-1} modulo 998244353.

Sample Input 1

1 1

Sample Output 1

499122178

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 3 / 2.
Thus, P = 3, Q = 2, Q^{-1} = 499122177 (modulo 998244353), and P \times Q^{-1} = 499122178 (again, modulo 998244353).

Sample Input 2

2 2

Sample Output 2

831870297

The expected number of your correct answers is 17 / 6.

Sample Input 3

3 4

Sample Output 3

770074220

The expected number of your correct answers is 169 / 35.

Sample Input 4

10 10

Sample Output 4

208827570

Sample Input 5

42 23

Sample Output 5

362936761



Sample input:
1 1


Sample output:
499122178

Submission status: Wrong Answer

Solution code:
def cmb(n, r):
    if n - r < r: r = n - r
    if r == 0: return 1
    if r == 1: return n

    numerator = [n - r + k + 1 for k in range(r)]
    denominator = [k + 1 for k in range(r)]

    for p in range(2,r+1):
        pivot = denominator[p - 1]
        if pivot > 1:
            offset = (n - r) % p
            for k in range(p-1,r,p):
                numerator[k - offset] /= pivot
                denominator[k] /= pivot

    result = 1
    for k in range(r):
        if numerator[k] > 1:
            result *= int(numerator[k])
            result%=mod

    return result
mod=998244353
n,M=map(int,input().split())
n*=2
a=2**(n-1) + (n-1)* cmb(n-1, n//2)
b=cmb(n,n//2)
print((a*pow(b,-1,mod))%mod)

Submission status: Time Limit Exceeded

Solution code:
import sys
from fractions import Fraction
import json

sys.setrecursionlimit(500000)
memo = {}
mod = 998244353

def e(m, n):

    if m < n:
        m, n = n, m

    if (m,n) in memo:
        return memo[(m,n)]

    if m == 0:
        return n
    if n == 0:
        return m
    if m == n:
        memo[(m,n)] = e(m-1, n)+Fraction(1,2)
    else:
        memo[(m,n)] = (e(m-1, n)+1)*Fraction(m,m+n)+e(m, n-1)*Fraction(n, m+n)
    return memo[(m,n)]


def main():
    n, m = list(map(int, input().split()))
    res = e(n, m)
    p = res.numerator
    q = res.denominator
    print((p*pow(q, mod-2, mod))%mod)


if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:
M=8**7
m,f,g,i=M*476+1,[j:=1],[k:=1]*M,0
while i<M:i+=1;f+=f[-1]*i%m,
g+=pow(f[-1],m-2,m),
while i:g[i-1]=g[i]*i%m;i-=1
A,B=map(int,input().split())
if A<B:A,B=B,A
while j<=B:i+=k*f[A+B-j]*g[B-j];k=k*2%m;j+=1
print((i*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
M=8**7
m,f,g,i=M*476+1,[j:=1],[k:=1]*M,1
while i<M:f+=f[-1]*i%m,;i+=1
g+=pow(M*f[-1],m-2,m),
while i:g[i-1]=g[i]*i%m;i-=1
A,B=map(int,input().split())
if A<B:A,B=B,A
while j<=B:i+=k*f[A+B-j]*g[B-j];k=k*2%m;j+=1
print((i*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
M=8**7
m,f,g,i,a,b=M*476+1,[1],[1]*M,1,0,1
while i<M:f+=f[-1]*i%m,;i+=1
g[-1]=pow(f[-1],m-2,m)
while i>1:i-=1;g[i-1]=g[i]*i%m
A,B=map(int,input().split())
if A<B:A,B=B,A
while i<=B:a+=b*f[A+B-i]*g[B-i];b=b*2%m;i+=1
print((a*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
M=8**7
m=M*476+1
f=[1]*M
g=[1]*M
h=[1]*M
i=2
while i<M:f[i],h[i]=f[i-1]*i%m,-h[m%i]*(m//i)%m;g[i]=g[i-1]*h[i]%m;i+=1
a=0
b=j=1
A,B=map(int,input().split())
if A<B:A,B=B,A
while j<=B:a+=b*f[A+B-j]*g[B-j];b=b*2%m;j+=1
print((a*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
M=8**7
m=M*476+1
f=[1]*M
g=[1]*M
h=[1]*M
i=2
while i<M:f[i]=f[i-1]*i%m;h[i]=-h[m%i]*(m//i)%m;g[i]=g[i-1]*h[i]%m;i+=1
a=0
b=j=1
A,B=map(int,input().split())
if A<B:A,B=B,A
while j<=B:a+=b*f[A+B-j]*g[B-j];b=b*2%m;j+=1
print((a*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
M=8**7
m=M*476+1
f=[1]*M
g=[1]*M
h=[1]*M
for i in range(2,M):
	f[i]=f[i-1]*i%m
	h[i]=-h[m%i]*(m//i)%m
	g[i]=g[i-1]*h[i]%m
A,B=map(int,input().split())
if A<B:A,B=B,A
a=0
b=1
for j in range(1,B+1):
	a+=b*f[A+B-j]*g[B-j]
	b=b*2%m
print((a*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
m=998244353
M=6**8
f=[1]*M
g=[1]*M
h=[1]*M
h[0]=0
for i in range(2,M):
	f[i]=f[i-1]*i%m
	h[i]=m-h[m%i]*(m//i)%m
	g[i]=g[i-1]*h[i]%m
A,B=map(int,input().split())
if A<B:A,B=B,A
a=0
b=1
for j in range(1,B+1):
	a+=b*f[A+B-j]*g[B-j]
	b=b*2%m
print((a*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
m=998244353
M=1<<20
f=[1]*M
g=[1]*M
h=[1]*M
h[0]=0
for i in range(2,M):
	f[i]=f[i-1]*i%m
	h[i]=m-h[m%i]*(m//i)%m
	g[i]=g[i-1]*h[i]%m
A,B=map(int,input().split())
if A<B:A,B=B,A
a=0
b=1
for j in range(1,B+1):
	a+=b*f[A+B-j]*g[B-j]*g[A]
	b=b*2%m
print((a*f[A]*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
m=998244353
M=1<<20
f=[0]*M
g=[0]*M
h=[0]*M
f[0]=g[0]=f[1]=g[1]=h[1]=1
for i in range(2,M):
	f[i]=f[i-1]*i%m
	h[i]=m-h[m%i]*(m//i)%m
	g[i]=g[i-1]*h[i]%m
A,B=map(int,input().split())
if A<B:A,B=B,A
a=0
b=1
for j in range(1,B+1):
	a+=b*f[A+B-j]*g[B-j]*g[A]
	b=b*2%m
print((a*f[A]*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
m=998244353
M=1<<20
f=[0]*M
g=[0]*M
h=[0]*M
f[0]=g[0]=f[1]=g[1]=h[1]=1
for i in range(2,M):
	f[i]=f[i-1]*i%m
	h[i]=m-h[m%i]*(m//i)%m
	g[i]=g[i-1]*h[i]%m
A,B=map(int,input().split())
if A<B:
	A,B=B,A
b=1
a=0
for j in range(1,B+1):
	a+=b*f[A+B-j]*g[B-j]*g[A]
	b=b*2%m
print((a*f[A]*f[B]*g[A+B]+A)%m)

Submission status: Accepted

Solution code:
m=998244353
M=1000001
f=[0]*M
g=[0]*M
h=[0]*M
f[0]=g[0]=f[1]=g[1]=h[1]=1
for i in range(2,M):
	f[i]=f[i-1]*i%m
	h[i]=m-h[m%i]*(m//i)%m
	g[i]=g[i-1]*h[i]%m
A,B=map(int,input().split())
if A<B:
	A,B=B,A
b=1
a=0
for j in range(1,B+1):
	a+=b*f[A+B-j]*g[B-j]*g[A]
	b=b*2%m
a=a*f[A]*f[B]*g[A+B]+A
print(a%m)

Submission status: Accepted

Solution code:
m=998244353
M=1000001
f=[0]*M
g=[0]*M
h=[0]*M
f[0]=g[0]=f[1]=g[1]=h[1]=1
for i in range(2,M):
	f[i]=f[i-1]*i%m
	h[i]=m-h[m%i]*(m//i)%m
	g[i]=g[i-1]*h[i]%m
A,B=map(int,input().split())
if A<B:
	A,B=B,A
b=1
ans=0
for j in range(1,B+1):
	ans+=b*f[A+B-j]*g[B-j]*g[A]%m
	b=b*2%m
ans=ans*f[A]*f[B]*g[A+B]+A
print(ans%m)

Submission status: Runtime Error

Solution code:
import math
import sys
from functools import lru_cache
mod=998244353
sys.setrecursionlimit(1000000)
@lru_cache(maxsize=1000000)
def ans(p,q,wa):
  if p<q:
    p,q=q,p
  if q==0:
    return wa+p
  else:
    return (ans(p-1,q,wa+1)+ans(p,q-1,wa))%mod
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod
N,M=map(int,input().split())
p = N+M
g1 = [1, 1] # 元テーブル
g2 = [1, 1] #逆元テーブル
inverse = [0, 1] #逆元テーブル計算用テーブル
for i in range( 2, p + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )
a=cmb(N+M,M,mod)
print(ans(N,M,0)*pow(a,-1,mod)%mod)

Submission status: Runtime Error

Solution code:
import math
import sys
from functools import lru_cache
sys.setrecursionlimit(1000000)
@lru_cache(maxsize=1000)
def ans(a,b,p,q,wa):
  if p<q:
    p,q=q,p
  if q==0:
    return a*(wa+p)*pow(b,-1,mod)%mod
  else:
    return (ans(a*p%mod,b*(p+q)%mod,p-1,q,wa+1)+ans(a*q%mod,b*(p+q)%mod,p,q-1,wa))%mod
mod=998244353
N,M=map(int,input().split())
print(ans(1,1,N,M,0))

Submission status: Runtime Error

Solution code:
import math
def tasizan(a,b,c,d):
  return a*d+b*c,b*d
mod=998244353
#N=M
N,M=map(int,input().split())
def ans(a,b,p,q,wa):
  if p<q:
    p,q=q,p
  if q==0:
    return a*(wa+p),b
  else:
    s,t=ans(a*p,b*(p+q),p-1,q,wa+1)
    c,d=ans(a*q,b*(p+q),p,q-1,wa)
    return tasizan(s,t,c,d)
a,b=ans(1,1,N,M,0)
p=math.gcd(a,b)
a,b=a//p,b//p
print(a*pow(b,-1,mod)%mod)
#これだめならDPでいく

Submission status: Runtime Error

Solution code:
M=8**7
m,f,g,i=M*476+1,[j:=1],[k:=1]*M,0
while i<M:i+=1;f+=f[-1]*i%m,
g+=pow(f[-1],m-2,m),
while i:g[i-1]=g[i]*i%m;i-=1
A,B=map(int,input().split())
if A<B:A,B=B,A
while j<=B:i+=k*f[A+B-j]*g[B-j];k=k*2%m;j+=1
print((i*f[B]*g[A+B]+A)%m))

Submission status: Runtime Error

Solution code:
import sys
from fractions import Fraction
import json

sys.setrecursionlimit(10000)
memo = {}

def e(m, n):

    if m < n:
        m, n = n, m

    if (m,n) in memo:
        return memo[(m,n)]

    if m == 0:
        return n
    if n == 0:
        return m
    if m == n:
        memo[(m,n)] = e(m-1, n)+Fraction(1,2)
    else:
        memo[(m,n)] = (e(m-1, n)+1)*Fraction(m,m+n)+e(m, n-1)*Fraction(n, m+n)
    return memo[(m,n)]


def main():
    n, m = list(map(int, input().split()))
    mod = 998244353
    res = e(n, m)
    p = res.numerator
    q = res.denominator
    print((p*pow(q, mod-2, mod))%mod)


if __name__ == '__main__':
    main()

Submission status: Runtime Error

Solution code:
from fractions import *

def E(n,m):
	if not m:return Fraction(n,1)
	if(n<m):return E(m,n)
	return Fraction(n*E(n-1,m)+m*E(n,m-1)+max(n,m),n+m)

def sqr_mod(a,m):return a*a%m

def pow_mod(a,n,m):
	if not n:return 1
	if n%2:return sqr_mod(pow_mod(a,n>>1,m),m)*a%m
	return sqr_mod(pow_mod(a,n>>1,m),m)

n,m=map(int,input().split())
x,mod=E(n,m),998244353
print(x.numerator*pow_mod(x.denominator,mod-2,mod)%mod)

Submission status: Runtime Error

Solution code:
from fractions import Fraction

memo = {}

def e(m, n):

    if m < n:
        m, n = n, m

    if (m,n) in memo:
        return memo[(m,n)]

    if m == 0:
        return n
    if n == 0:
        return m
    if m == n:
        memo[(m,n)] = e(m-1, n)+Fraction(1,2)
    else:
        memo[(m,n)] = (e(m-1, n)+1)*Fraction(m,m+n)+e(m, n-1)*Fraction(n, m+n)
    return memo[(m,n)]


def main():
    n, m = list(map(int, input().split()))
    mod = 998244353
    res = e(n, m)
    p = res.numerator
    q = res.denominator
    print((p*pow(q, mod-2, mod))%mod)


if __name__ == '__main__':
    main()