Difference between revisions of "2022 USAJMO Problems/Problem 1"
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Revision as of 18:02, 6 October 2023
Problem
For which positive integers does there exist an infinite arithmetic sequence of integers and an infinite geometric sequence of integers satisfying the following properties?
is divisible by for all integers ;
is not divisible by .
Solution 1
We claim that satisfies the given conditions if and only if is squareful.
To begin, we let the common difference be and the common ratio be . Then, rewriting the conditions modulo gives:
Condition holds iff no consecutive terms in are equivalent modulo , which is the same thing as never having consecutive, equal, terms, in . By Condition , this is also the same as never having equal, consecutive, terms in :
Also, Condition holds iff
Whee! Restating, , and the conditions and hold if and only if is squareful.
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See Also
2022 USAJMO (Problems • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.