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Combinatorics Difficulty 4.5 AIME Find the answer United States

Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, “I can’t tell who has the larger number.” Then Bob says, “I know who has the larger number.” Alice says, “You do? Is your number prime?” Bob replies, “Yes.” Alice says, “In that case, if I multiply your number by $100$ and add my number, the result is a perfect square.” What is the sum of the two numbers drawn from the hat?

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Solution

Based on Alice's first statement, Bob can deduce that her number is not $1$ or $40$ . Bob says that he knows who has the larger number, which implies that his number must be $1$ , $2$ , $39$ , or $40$ . The number $2$ is the only prime among them, which tells Alice that Bob's number is $2$ . Alice then says that her number added to $200$ is a perfect square. The only perfect square between $201$ and $240$ is $15^2 = 225$ , so Alice's number is $25$ and the sum of the two numbers is $25 + 2 = 27$ .

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