Difference between revisions of "2021 Fall AMC 10B Problems/Problem 10"
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+ | ==Problem== | ||
+ | Fourty slips of paper numbered <math>1</math> to <math>40</math> 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 <math>100</math> and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat? | ||
+ | |||
+ | <math>\textbf{(A) }27\qquad\textbf{(B) }37\qquad\textbf{(C) }47\qquad\textbf{(D) }57\qquad\textbf{(E) }67</math> | ||
+ | |||
+ | ==Solution== | ||
+ | Because Alice doesn't know who has the larger number, she doesn't have <math>1.</math> Because Alice says that she doesn't know who has the larger number, Bob knows that she doesn't have <math>1.</math> 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 <math>2</math>. Thus, the perfect square is in the <math>200's.</math> The only perfect square is <math>225.</math> Thus, Alice's number is <math>25.</math> The sum of Alice's and Bob's number is <math>2+25 = 27.</math> Thus the answer is <math>\boxed{(\textbf{A}.)}.</math> | ||
+ | |||
+ | ~NH14 | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2021 Fall|ab=B|num-a=11|num-b=9}} | {{AMC10 box|year=2021 Fall|ab=B|num-a=11|num-b=9}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 02:28, 23 November 2021
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
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.