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.