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

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.

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 \( N \).

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.

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.

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)!}}.

\]

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 \( S_p \).

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.

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 \( \sqrt 2 \).