2022 AMC 10A Problems/Problem 17
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Problem
How many three-digit positive integers are there whose nonzero digits and satisfy (The bar indicates repetition, thus is 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 | |
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