Difference between revisions of "2022 AMC 10A Problems/Problem 17"
MRENTHUSIASM (talk | contribs) (→Solution) |
(→Video Solution 2 (HOW TO THINK CREATIVELY)) |
||
Line 37: | Line 37: | ||
==Video Solution 1== | ==Video Solution 1== | ||
https://www.youtube.com/watch?v=YAazoVATYQA&list=PLmpPPbOoDfgj5BlPtEAGcB7BR_UA5FgFj&index=4 | https://www.youtube.com/watch?v=YAazoVATYQA&list=PLmpPPbOoDfgj5BlPtEAGcB7BR_UA5FgFj&index=4 | ||
− | |||
− | |||
− | |||
− | |||
− | |||
== See Also == | == See Also == |
Revision as of 15:02, 14 August 2023
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
Remark
One way to solve the Diophantine Equation, is by taking , from which the equation becomes so either or WLOG .
Video Solution 1
https://www.youtube.com/watch?v=YAazoVATYQA&list=PLmpPPbOoDfgj5BlPtEAGcB7BR_UA5FgFj&index=4
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.