# 2013-11 Integer coefficient complex-valued polynomials

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

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# Solution: 2013-10 Mean and variance of random variable

Let random variables $$\{ X_r : r \geq 1 \}$$ be independent and uniformly distributed on $$[0, 1]$$. Let $$0 < x < 1$$ and define a random variable $N = \min \{ n \geq 1 : X_1 + X_2 + \cdots + X_n > x \}.$
Find the mean and variance of $$N$$.

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

Similar solutions were also submitted by 라준현(08학번, +3), 서기원(09학번, +3), 김범수(10학번, +3), 황성호(13학번, +3), 어수강(서울대, +3), 이시우(POSTECH, +3), Fardad Pouran(Sharif University of Tech, Iran, +3), 양지훈(10학번, +2), 이정민(서울대, +2). Thank you for your participation.

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# 2013-10 Mean and variance of random variable

Let random variables $$\{ X_r : r \geq 1 \}$$ be independent and uniformly distributed on $$[0, 1]$$. Let $$0 < x < 1$$ and define a random variable $N = \min \{ n \geq 1 : X_1 + X_2 + \cdots + X_n > x \}.$
Find the mean and variance of $$N$$.

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# Solution: 2013-09 Inequality for a sequence

Let $$N > 1000$$ be an integer. Define a sequence $$A_n$$ by
$A_0 = 1, \, A_1 = 0, \, A_{2k+1} = \frac{2k}{2k+1} A_{2k} + \frac{1}{2k+1} A_{2k-1}, \, A_{2k} = \frac{2k-1}{2k} \frac{A_{2k-1}}{N} + \frac{1}{2k} A_{2k-2}.$
Show that the following inequality holds for any integer $$k$$ with $$1 \leq k \leq (1/2) N^{1/3}$$.
$A_{2k-2} \leq \frac{1}{\sqrt{(2k-2)!}}.$

The best solution was submitted by 어수강, 서울대학교 석사과정. Congratulations!

An alternative solution was submitted by 라준현(08학번, +3). Thank you for your participation.

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# Solution: 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 $$S_p$$.

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

Other solutions were submitted by 라준현(08학번, +3), 어수강(서울대, +3). Thank you for your participation.

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# 2013-09 Inequality for a sequence

Let $$N > 1000$$ be an integer. Define a sequence $$A_n$$ by
$A_0 = 1, \, A_1 = 0, \, A_{2k+1} = \frac{2k}{2k+1} A_{2k} + \frac{1}{2k+1} A_{2k-1}, \, A_{2k} = \frac{2k-1}{2k} \frac{A_{2k-1}}{N} + \frac{1}{2k} A_{2k-2}.$
Show that the following inequality holds for any integer $$k$$ with $$1 \leq k \leq (1/2) N^{1/3}$$.
$A_{2k-2} \leq \frac{1}{\sqrt{(2k-2)!}}.$

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# 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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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 $$\sqrt 2$$.