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Difference between revisions of "2021 CMC 12A Problems/Problem 4"

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{{duplicate|[[2021 CMC 12A Problems | 2021 CMC 12A #4]] and [[2021 CMC 10A Problems | 2021 CMC 10A #5]]}}
 
{{duplicate|[[2021 CMC 12A Problems | 2021 CMC 12A #4]] and [[2021 CMC 10A Problems | 2021 CMC 10A #5]]}}
 
==Problem==
 
==Problem==
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There exists a positive integer <math>N</math> such that <cmath>\frac{\tfrac{1}{999}+\tfrac{1}{1001}}{2}=\frac{1}{N}.</cmath> What is the sum of the digits of <math>N</math>?
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There exists a positive integer <math>N</math> such that <cmath>\frac{\tfrac{1}{999}+\tfrac{1}{1001}}{2}=\frac{1}{1000}+\frac{1}{N}.</cmath> What is the sum of the digits of <math>N</math>?
  
 
<math>\textbf{(A) } 36\qquad\textbf{(B) } 45\qquad\textbf{(C) } 54\qquad\textbf{(D) } 63\qquad\textbf{(E) } 72\qquad</math>
 
<math>\textbf{(A) } 36\qquad\textbf{(B) } 45\qquad\textbf{(C) } 54\qquad\textbf{(D) } 63\qquad\textbf{(E) } 72\qquad</math>
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{{CMC12 box|year=2021|ab=A|num-b=3|num-a=5}}
 
{{CMC12 box|year=2021|ab=A|num-b=3|num-a=5}}
 
{{CMC10 box|year=2021|ab=A|num-b=4|num-a=6}}
 
{{CMC10 box|year=2021|ab=A|num-b=4|num-a=6}}
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{{MAC Notice}}

Latest revision as of 23:41, 24 March 2023

The following problem is from both the 2021 CMC 12A #4 and 2021 CMC 10A #5, so both problems redirect to this page.

Problem

There exists a positive integer $N$ such that \[\frac{\tfrac{1}{999}+\tfrac{1}{1001}}{2}=\frac{1}{1000}+\frac{1}{N}.\] What is the sum of the digits of $N$?

$\textbf{(A) } 36\qquad\textbf{(B) } 45\qquad\textbf{(C) } 54\qquad\textbf{(D) } 63\qquad\textbf{(E) } 72\qquad$

Solution

Let $a=1000$. Then $\frac{\tfrac{1}{a-1}+\frac{1}{a+1}}{2}-\frac{1}{a}=\frac{1}{N}$ or $\frac{1}{a^{3}-a}=\frac{1}{N}$, so $N=a^{3}-a=1000^{3}-1000=999,999,000$ whose digits sum to $\boxed{\textbf{(C) } 54}$.

Video Solution

https://youtu.be/H4uIpLB-6Jo

~Punxsutawney Phil

See also

2021 CMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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 CMC 12 Problems and Solutions
2021 CMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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 CMC 10 Problems and Solutions

The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions. AMC logo.png

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