Let \(A, B\) are \(N \times N \) complex matrices satisfying \( rank(AB – BA) = 1 \). Prove that \( (AB – BA)^2 = 0 \).

The best solution was submitted by 김호진. Congratulations!

Similar solutions were submitted by 강동엽(+3), 김범수(+3), 김홍규(+3), 박민재(+3), 박지민(+3), 박훈민(+3), 안가람(+3), 어수강(+3), 엄문용(+3), 유찬진(+3), 이성회(+3), 이시우(+3), 이주호(+3), 장경석(+3), 전한솔(+3), 정동욱(+3), 정성진(+3), 정종헌(+3), 정우석(+3), 진우영(+3), Fardad Pouran(+3). Thank you for your participation.

Find all continuous functions \(f : \mathbb{R} \to \mathbb{R}\) satisfying
\[
f(x) = f(x^2 + \frac{x}{3} + \frac{1}{9} )
\]
for all \( x \in \mathbb{R} \).

The best solution was submitted by 강동엽. Congratulations!

Similar solutions were submitted by 김기현(+3), 김범수(+3), 김정섭(+3), 김호진(+3), 김홍규(+3), 박민재(+3), 박지민(+3), 박훈민(+3), 어수강(+3), 엄문용(+3), 윤성철(+3), 이명재(+3), 이성회(+3), 이시우(+3), 이주호(+3), 장경석(+3), 전한솔(+3), 정동욱(+3), 정성진(+3), 정종헌(+3), 조정휘(+3), 진우영(+3), 안가람(+2), 박경호(+2), 정우석(+2). Thank you for your participation.

Remark 1. As written in the rules, please submit the solution by 12PM on Wednesday. Any solution submitted after 12PM will not be graded.
Remark 2. Please write your name in the solution (not just in the email).

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.

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.