Difference between revisions of "2022 AMC 10A Problems/Problem 17"
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<math>\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14</math> | <math>\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14</math> | ||
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We rewrite the given equation, then rearrange: | We rewrite the given equation, then rearrange: | ||
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
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== See Also == | == See Also == |
Revision as of 22:52, 4 December 2022
Problem
How many three-digit positive integers are there whose nonzero digits
and
satisfy
(The bar indicates repetition, thus
in the infinite repeating decimal
)
Solution
We rewrite the given equation, then rearrange:
Now, this problem is equivalent to counting the ordered triples
that satisfies the equation.
Clearly, the ordered triples
are solutions to this equation.
The expression has the same value when:
increases by
as
decreases by
decreases by
as
increases by
We find more solutions from the
solutions above:
Note that all solutions are symmetric about
Together, we have ordered triples
~MRENTHUSIASM
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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.