2021 Fall AMC 10B Problem 10
Problem 10 of 25EasierNumber Theory
Forty slips of paper numbered to 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 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
If Alice had drawn or she would know who had the larger number. Her first statement tells Bob that Alice has neither nor
Bob can then know who has the larger number only if his number is or Since Bob says his number is prime, his number must be
Now plus Alice’s number is a square between and The only square in that interval is so Alice’s number is The sum is
Thus, the answer is A .