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

The best solution was submitted by 황성호, 13학번. Congratulations!

Another solution was submitted by 라준현(08학번, +3). 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 \( N \).

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 \( N > 1000 \) 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.

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.

Let \( f : [0, 1] \to \mathbb{R} \) be a continuously differentiable function with \( f(0) = 0 \) and \( 0 < f'(x) \leq 1 \). Prove that \[ \left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx. \]

The best solution was submitted by 박훈민, 13학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +3), 서기원(09학번, +3), 김범수(10학번, +3), 황성호(13학번, +3), 홍혁표(13학번, +3), 김준(13학번, +3), 전한솔(고려대 13학번, +3), 이시우(POSTECH 13학번, +3), 한대진(신현여중 교사, +3). Thank you for your participation.

Let \[ F(x) = \sum_{n=1}^{1000} \cos (n^{3/2} x). \]
Prove that \( F \) has at least \( 80 \) zeros in the interval \( (0, 2013) \).

The best solution was submitted by 황성호, 13학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +2). Thank you for your participation. Sincere apology for the error in the first version last Friday.

Let \( H \) be an \( N \times N \) real symmetric matrix. Suppose that \( |H_{kk}| < 1 \) for \( 1 \leq k \leq N \). Prove that, if \( |H_{ij}| > 4 \) for some \( i, j \), then the largest eigenvalue of \( H \) is larger than \( 3 \).

The best solution was submitted by 김범수, 10학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 서기원(09학번, +3), 김호진(09학번, +3), 김범수(10학번, +3), 박훈민(13학번, +3), 노수현(13학번, +2). Thank you for your participation.

Let \( t \) be a positive real number and \( m \) be a positive integer. Show that if both \( \cosh \, mt \) and \( \cosh \, (m+1)t \) are rational then \( \cosh \, t \) is also rational.

The best solution was submitted by 홍혁표, 13학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 서기원(09학번, +3), 김호진(09학번, +3), 김범수(10학번, +3), 박지민(12학번, +3), 김정민(12학번, +2), 양지훈(10학번, +2), 황성호(13학번, +2). Thank you for your participation.

Let \( \mathbb{Z}^+ \) be the set of positive integers. Suppose that \( f : \mathbb{Z}^+ \to \mathbb{Z}^+ \) satisfies the following conditions.

i) \( f(f(x)) = 5x \).

ii) If \( m \geq n \), then \( f(m) \geq f(n) \).

iii) \( f(1) \neq 2 \).

Find \( f(256) \).

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

Similar solutions were also submitted by 황성호(13학번, +3), 양지훈(10학번, +3), 홍혁표(13학번, +3), 김준(13학번, +3), 서기원(09학번, +3), 이주호(12학번, +3), 박훈민(13학번, +3), 송유신(10학번, +3), 임현진(10학번, +3), 라준현(08학번, +3), 김정민(12학번, +3), 박지민(12학번, +3), 김태호(11학번, +3), 김범수(10학번, +3), 전한솔(고려대 13학번, +3), 어수강(서울대 석사과정, +3), 이시우(POSTECH 13학번, +3), 정우석(서강대 11학번, +3), 윤성철(홍익대 09학번, +3), 김재호(하나고, +3), 이정준(08학번, +2). Thank you for your participation.