2021 Fall AMC 10B Problems/Problem 10
Problem
Fourty 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?
Solution
Because Alice doesn't know who has the larger number, she doesn't have 
Because Alice says that she doesn't know who has the larger number, Bob knows that she doesn't have But Bob knows who has the larger number, this implies that Bob has the smallest possible number. Because Bob's number is prime, Bob's number is 
. Thus, the perfect square is in the 
The only perfect square is 
Thus, Alice's number is 
The sum of Alice's and Bob's number is 
Thus the answer is 
~NH14
Video Solution by Interstigation
https://youtu.be/p9_RH4s-kBA?t=1524
See Also
| 2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
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| All AMC 10 Problems and Solutions | ||
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