# 2013-08 Minimum of a set involving polynomials with integer coefficients

Let $$p$$ be a prime number. Let $$S_p$$ be the set of all positive integers $$n$$ satisfying
$x^n – 1 = (x^p – x + 1) f(x) + p g(x)$
for some polynomials $$f$$ and $$g$$ with integer coefficients. Find all $$p$$ for which $$p^p -1$$ is the minimum of $$S_p$$.

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# Solution: 2013-07 Maximum number of points

Consider the unit sphere in $$\mathbb{R}^n$$. Find the maximum number of points on the sphere such that the (Euclidean) distance between any two of these points is larger than $$\sqrt 2$$.

The best solution was submitted by 라준현, 08학번. Congratulations!

Other solutions were submitted by 서기원(09학번, +3), 황성호(13학번, +3), 김범수(10학번, +3), 전한솔(고려대, +3), 홍혁표(13학번, +2), 어수강(서울대, +2). Thank you for your participation.

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