2021 CMC 12A Problems/Problem 4

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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 (ProblemsAnswer KeyResources)
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 (ProblemsAnswer KeyResources)
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

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