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.

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

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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.

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