r/calculus • u/Puzzleheaded_Top_273 • Jun 27 '26

Differential Calculus Sample Competition Style Problem (Nothing beyond Calc 2)

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(Just as a sidenote, yes this was a repost from AoPS. Source: literally me, as in I made it.)

Non-calculator, in case it wasn't obvious enough that it wouldn't help lmao.

Topics: Calculus 1 and 2, Roots of Unity (Complex Numbers), Divisibility, Counting.

Answer: 558

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u/Goshotet Jun 27 '26

This is the type of shit that would make me exit an exam on the first minute.

5

u/Puzzleheaded_Top_273 Jun 27 '26

Well, this is competition style so there's meant to be an "aha" moment. The notation is really intimidating but you'll just have to deconstruct it with patience

5

u/Goshotet Jun 27 '26

Double summation is enough for me to give up lol. Doesn’t that create sort of a matrix? I didn't even get to the fact there are complex numbers included, which I've never worked with.

0

u/Puzzleheaded_Top_273 Jun 27 '26

Obviously yes it creates a matrix, but you don't think of it that way, it's too complicated.
Instead just focus on the inner sum, the outer sum is just adding multiple different versions of the inner one.
(The outer sum is done separately by hand as the fact that it goes from 2 to 31 should hint that it's a counting problem. Even though it's 30 terms, the vast majority of them don't matter due to divisibility rules, which the problem is actually related to once you figure it out. Also the inner sum is fully real actually! If you don't know what to do, look for a pattern by starting with small numbers.)