Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 Fall AMC 10B Problems/Problem 10"

(Added a short sidenote)
m
Line 37: Line 37:
  
 
~savannahsolver
 
~savannahsolver
 +
==Video Solution by TheBeautyofMath==
 +
https://youtu.be/RyN-fKNtd3A?t=1474
 +
 +
~IceMatrix
 
==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 00:01, 30 December 2022

Problem

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?

$\textbf{(A) }27\qquad\textbf{(B) }37\qquad\textbf{(C) }47\qquad\textbf{(D) }57\qquad\textbf{(E) }67$

Solution

Denote by $A$ and $B$ the numbers drawn by Alice and Bob, respectively.

Alice's sentence ``I can't tell who has the larger number. implies $A \in \left\{ 2 , \cdots , 39 \right\}$.

Bob's sentence ``I know who has the larger number. implies $B \in \left\{ 1 , 2 , 39, 40 \right\}$.

Their subsequent conversation that $B$ is prime implies $B = 2$.

Then, Alice's next sentence ``In that case, if I multiply your number by 100 and add my number, the result is a perfect square. implies $200 + A$ is a perfect square. Hence, $A = 25$.

Therefore, the answer is $\boxed{\textbf{(A) }27}$.

~Steven Chen (www.professorchenedu.com)

Sidenote

Note that Bob's statement (that he knows who won/lost) comes after Alice tells him that she doesn't know who won. Since Alice doesn't know who won, Bob knows that she didn't get draw a $1$ (or a $40$), which tells him that his $2$ must be lower than Alice's number.

~jd9

Video Solution by Interstigation

https://youtu.be/p9_RH4s-kBA?t=1524

Video Solution

https://youtu.be/vB3R4b-X3yA

~Education, the Study of Everything

Video Solution by WhyMath

https://youtu.be/j-AZyR78-Ns

~savannahsolver

Video Solution by TheBeautyofMath

https://youtu.be/RyN-fKNtd3A?t=1474

~IceMatrix

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_Fall_AMC_10B_Problems/Problem_10&oldid=185226"