# Concluding Spring 2013

The top 5 participants of the semester are:

• 1st: 라준현 (08학번): 38 points
• 2nd: 서기원 (09학번): 29 points
• T-3rd: 김호진 (09학번): 25 points
• T-3rd: 황성호 (13학번): 25 points
• 5th: 김범수 (10학번): 19 points

Hearty congratulations to the prize winners! The prize ceremony will be held on Jun. 19 (Wed.) at 2PM.

We thank all of the participants for the nice solutions and your intereset you showed for POW. We hope to see you next semester with even better problems.

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# Solution: 2013-12 Equilateral triangle in R^n

Let $$A = \{ (a_1, a_2, \cdots, a_n : a_i = \pm 1 \, (i = 1, 2, \cdots, n) \} \subset \mathbb{R}^n$$. Prove that, for any $$X \subset A$$ with $$|X| > 2^{n+1}/n$$, there exist three distinct points in $$X$$ that are the vertices of an equilateral triangle.

The best solution was submitted by 서기원, 09학번. Congratulations!

Similar solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +3), 황성호(13학번, +3), 박정현(일반, +3), 정요한(서울시립대, +3). Thank you for your participation.

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# 2013-12 Equilateral triangle in R^n

Let $$A = \{ (a_1, a_2, \cdots, a_n : a_i = \pm 1 \, (i = 1, 2, \cdots, n) \} \subset \mathbb{R}^n$$. Prove that, for any $$X \subset A$$ with $$|X| > 2^{n+1}/n$$, there exist three distinct points in $$X$$ that are the vertices of an equilateral triangle.

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Determine all polynomials $$P(z)$$ with integer coefficients such that, for any complex number $$z$$ with $$|z| = 1$$, $$| P(z) | \leq 2$$.