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Omni-MATH / Paul and Sara are playing a game with integers on a whiteboard, with Paul going…

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problem

Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even. Initially, there are 2021 integers on the board, each one sampled uniformly at random from the set \{0,1,2,3, \ldots, 2021\}. Assuming both players play optimally, the probability that Paul wins is mn\frac{m}{n}, where m,nm, n are positive integers and gcd(m,n)=1\operatorname{gcd}(m, n)=1. Find the remainder when m+nm+n is divided by 1000.
Plain-text mathematical notation (without MathML)
Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even. Initially, there are 2021 integers on the board, each one sampled uniformly at random from the set \{0,1,2,3, \ldots, 2021\}. Assuming both players play optimally, the probability that Paul wins is (m)/(n), where m,n are positive integers and gcd(m,n)=1. Find the remainder when m+n is divided by 1000.
Original LaTeX notation
Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even. Initially, there are 2021 integers on the board, each one sampled uniformly at random from the set \{0,1,2,3, \ldots, 2021\}. Assuming both players play optimally, the probability that Paul wins is $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find the remainder when $m+n$ is divided by 1000.

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Source and history

task_id
0e04f1aa-5e1c-5b83-98b0-2ad3f90193a8
task_key
test--0e04f1aa-5e1c-5b83-98b0-2ad3f90193a8
task_revision_id
1
config / split
/ test

Official source

initial import