# Solution: 2024-10 Supremum

Find
$\sup \left[ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \left( \sum_{i=n}^{\infty} x_i^2 \right)^{1/2} \Big/ \sum_{i=1}^{\infty} x_i \right],$
where the supremum is taken over all monotone decreasing sequences of positive numbers $$(x_i)$$ such that $$\sum_{i=1}^{\infty} x_i < \infty$$.

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

There were incorrect solutions submitted.

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# Notice on POW 2024-05 and POW 2024-06

It is found that there is a flaw in POW 2024-05; some students showed that the collection of all Knotennullstelle numbers is not a discrete subset of $$\mathbb{C}$$. We again apologize for the inconvenience.

To acknowledge the students who reported the flaws in POW 2024-05 and POW 2024-06, we decided to give credits to 김준홍 (KAIST 수리과학과 20학번, +4) and 지은성 (KAIST 수리과학과 20학번, +3) for POW 2024-05 and Anar Rzayev (KAIST 전산학부 19학번, +4) for POW 2024-06.

Here is a “solution” of problem 2024-05.

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# Solution: 2024-04 Real random variable

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!

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.

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# Solution: 2024-03 Roots of complex derivative

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!

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.

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# Solution: 2023-14 Dividing polynomials

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!

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).

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