2022 USAJMO Problems/Problem 1
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 if 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 if
Whee! Restating, $(1),(2)\if (3),(4)$ (Error compiling LaTeX. Unknown error_msg), and the conditions and hold if and only if is squareful.
[will finish that step here]
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 |
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