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PrintFall 2021 AMC 10 B
United States 2021 counting and probability
Problem
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?
(A) (B) (C) (D) (E)
(A) (B) (C) (D) (E)
Solution
Based on Alice's first statement, Bob can deduce that her number is not or . Bob says that he knows who has the larger number, which implies that his number must be , , , or . The number is the only prime among them, which tells Alice that Bob's number is . Alice then says that her number added to is a perfect square. The only perfect square between and is , so Alice's number is and the sum of the two numbers is .
Final answer
A
Techniques
LogicPrime numbersIntegers