What I do in PE

by v_Enhance, Feb 24, 2012, 7:48 PM

This.
  1. (Balkan MO 1996) Let $p>5$ be a prime number and $X = \{p-n^2 \mid n \in \mathbb N, n^2 < p\}$. Prove that $X$ contains two distinct elements $x,y$ such that $x \neq 1$ and $x$ divides $y$.

  2. (BAMO 2004) Find all monic polynomials $f$ with integer coefficients such that
    • $f(0) = 2004$
    • $f(x)$ is irrational whenever $x$ is irrational.

  3. (USAMO 2010) There are $n$ students standing in a circle, one behind the other. The students have heights $h_1 < h_2 < \dots < h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k - 2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.

So I solved the first problem a year ago while walking to the locker room. I solved the second yesterday during PE while walking to a fence, and the third one today while walking to a basketball court.

I should walk more.
This post has been edited 3 times. Last edited by v_Enhance, Feb 25, 2012, 5:00 AM

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9 Comments

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Interesting. Einstein was known to take long walks and this supposedly helped him think.

by yugrey, Feb 24, 2012, 8:36 PM

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I solve a lot of math problems while in the bathroom. lolwut.

by dinoboy, Feb 24, 2012, 11:01 PM

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I think best when I walk too, but for some reason, I actually solve most hard math problems while in bed.

by 3333, Feb 25, 2012, 1:30 AM

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I solve most of my math problems on paper.

by dragon96, Feb 25, 2012, 2:41 AM

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This is why I like combo problems.

by abcak, Feb 25, 2012, 4:01 AM

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I've solved a couple while waiting to fall asleep; I seem to nail quite a few in the shower, because it fogs up the surface nearby, hence allowing me to write stuff.

And yeah combo is pretty tricky.

by v_Enhance, Feb 25, 2012, 4:30 AM

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lol, waiting to fall to sleep really increases my brain activity somehow as well..

by proglote, Feb 26, 2012, 1:52 AM

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Darn, nice. I can't do anything in my head.

by AwesomeToad, Feb 26, 2012, 9:30 PM

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I definitely think best in the middle of the night waiting to fall asleep.
The hardest geo problem I ever solved was one I solved in my head while trying to fall asleep at 11:30. (Trust me, it's pretty hard to do geo in your head)
The hardest inequality I ever solved was also in my head at 11:30 in my bed.
I solved this year's USAMTS Round 2 #5, one of the toughest combo problems I've done, at 2:30 in the morning randomly scribbling on paper on my pillow.

by pi37, Mar 4, 2012, 6:25 PM

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