Question

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?

$27$

$37$

$47$

$57$

$67$

Solution:

If Alice had drawn $1$ or $40$ , she would know who had the larger number. Her first statement tells Bob that Alice has neither $1$ nor $40$ .

Bob can then know who has the larger number only if his number is $1,2,39,$ or $40$ . Since Bob says his number is prime, his number must be $2$ .

Now $100\cdot2$ plus Alice's number is a square between $201$ and $240$ . The only square in that interval is $225$ , so Alice's number is $25$ . The sum is $2+25=27$ .

Thus, the answer is A .

2021 AMC 10B Fall Problem 10

Solution:

Problem 10 in Other Years