{"status": "success", "data": {"description_md": "**POTD August 14, 2024**\n\nPaul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul\u2019s turn, he can pick any two integers on the board and replace them with their product; whe it is Sara\u2019s 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 ${0, 1, 2, 3, . . . , 2021}$. Assuming both players play optimally, the probability that Paul wins is $\\frac{m}{n}$ where $m$ and $n$ are positive integers and $\\gcd (m, n) = 1$. Find the remainder when $m + n$ is divided by $1000$.", "description_html": "

POTD August 14, 2024

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; whe 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, . . . , 2021}. Assuming both players play optimally, the probability that Paul wins is \\frac{m}{n} where m and n are positive integers and \\gcd (m, n) = 1. Find the remainder when m + n is divided by 1000.

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