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 a perfect square.
To begin, we let the common difference of be and the common ratio of 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
Restating, , and the conditions and hold if and only if is a perfect square.
[will finish that step here]
Note: This shouldn't work since we see that m = 12 is a solution. Let the initials for both series by 1, then let the ratio be 7 and the common difference to be 6. We see multiplying by 7 mod 12 that the geometric sequence is alternating from 1 to 7 to 1 to 7 and so on, which is the same as adding 6. Therefore, this solution is wrong. My counter-conjecture is that all non square-free m (4, 8, 9, 16, 18, 25...) should all work, but I don't have a proof. However, if you edit the one above, you can see non square-free m will work. In order to construct a ratio, we could us (4) and find a square multiple of m, take the square root and add 1 to get the ratio. Let then or is not divisble by . If , this is false and this is not possible. But if it isn't, if isn't square free, then it should work.
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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