Suppose that \( a_1, a_2, \cdots \) are positive real numbers. Prove that
\[
\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} \leq e \sum_{n=1}^{\infty} a_n \,.
\]

The best solution was submitted by 정성진. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 이영민 (+3), 이종원 (+3), 장기정 (+3), 정성진 (+3), 조준영 (+3), 황성호 (+2). Incorrect solutions were submitted by K.S.J., L.S.C. (Some initials here might have been improperly chosen.)

Suppose that a function \( f:[0, 1] \to (0, \infty) \) satisfies that
\[ \int_0^1 f(x) dx = 1. \]
Prove the following inequality.
\[ \left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx. \]

The best solution was submitted by 정성진. Congratulations!

Similar solutions are submitted by 박민재(+3), 진우영(+3). Thank you for your participation.

Let \( x, y \) be real numbers satisfying \( y \geq x^2 + 1 \). Prove that there exists a bounded random variable \( Z \) such that
\[
E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y.
\]
Here, \( E \) denotes the expectation.

The best solution was submitted by 정성진. Congratulations!

Other solutions are submitted by 박민재(+3), 이주호(+3), 장경석(+3), 진우영(+3). Thank you for your participation.