Prove the following: There exists a bounded real random variable \( Z \) such that
\[
E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y
\]
if and only if \( y \geq x^2 + 1 \). (Here, \( E \) denotes the expectation.)

The best solution was submitted by 김준홍 (KAIST 수리과학과 20학번, +4). Congratulations!

Here is the best solution of problem 2024-04.

Other solutions were submitted by 신정연 (KAIST 수리과학과 21학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +2), 김찬우 (연세대학교 수학과 22학번, +2), 박상현 (고려대학교 수학과 20학번, +2), 이명규 (KAIST 전산학부 20학번, +2), 정영훈 (KAIST 새내기과정학부 24학번, +2). There were incorrect solutions submitted.

Let \(P(z) = z^3 + c_1 z^2 + c_2 z+ c_3\) be a complex polynomial in \(\mathbb{C}\). Its complex derivative is given by \(P’(z) = 3z^{2} +2c_1z+c_{2}.\) Assume that there exist two points a, b in the open unit disc of complex plane such that P(a) = P(b) =0. Show that  there is a point w belonging to the line segment joining a and b such that  \({\rm Re} (P’(w)) = 0\).

The best solution was submitted by 김준홍 (KAIST 수리과학과 20학번, +4). Congratulations!

Here is the best solution of problem 2024-03.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정 24학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 지은성 (KAIST 수리과학과 23학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +2), 박기윤 (KAIST 수리과학과 23학번, +2), 이명규 (KAIST 전산학부 20학번, +2), There were incorrect solutions submitted.

Let \(f(t)=(t^{pq}-1)(t-1) \) and \(g(t)=(t^{p}-1)(t^q-1) \) where \(p\) and \(q\) are relatively prime positive integers. Prove that \(\frac{f(t)}{g(t)}\) can be written as a polynomial where it has just \(1\) or \(-1\) as coefficients. (For example, when \(p=2\) and \(q=3\), we have that \(\frac{f(t)}{g(t)} = t^2-t+1\).)

The best solution was submitted by 김준홍 (KAIST 수리과학과 20학번, +4). Congratulations!

Here is the best solution of problem 2023-14.

Other solutions were submitted by 강지민 (세마고 3학년, +3), 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 전해구 (KAIST 기계공학과 졸업생, +3), 조현준 (KAIST 수리과학과 22학번, +4), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3). Muhammadfiruz Hasanov (+3).

Let \(\{x_1, x_2, \ldots, x_{21}\} = \{-10, -9, \ldots, -1, 0, 1, \ldots, 9, 10\}\). What is the largest possible value of \(x_1x_2x_3+x_4x_5x_6+\cdots + x_{19}x_{20}x_{21}\)?

The best solution was submitted by 김준홍 (KAIST 수리과학과 20학번, +4). Congratulations!

Here is the best solution of problem 2023-05.