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.