# Notice

There will be no POW this week due to 추석 (thanksgiving) break. POW will resume next week.

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# Solution: 2023-15 An inequality for complex polynomials

Let $$p(z), q(z)$$ and $$r(z)$$ be polynomials with complex coefficients in the complex plane. Suppose that $$|p(z)| + |q(z)| \leq |r(z)|$$ for every $$z$$. Show that there exist two complex numbers $$a,b$$ such that $$|a|^2 +|b|^2 =1$$ and $$a p(z) + bq(z) =0$$ for every $$z$$.

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Other solutions were submitted by 강지민 (세마고 3학년, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 신민서 (KAIST 수리과학과 20학번, +3), 여인영 (KAIST 물리학과 20학번, +3),이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 조현준 (KAIST 수리과학과 22학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 최민규 (한양대학교 의과대학 졸업생, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), Muhammadfiruz Hasanov (+3).

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# 2023-16 Zeros in a sequence

Define the sequence $$x_n$$ by $$x_1 = 0$$ and
$x_n = x_{\lfloor n/2 \rfloor} + (-1)^{n(n+1)/2}$
for $$n \geq 2$$. Find the number of $$n \leq 2023$$ such that $$x_n = 0$$.

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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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# 2023-15 An inequality for complex polynomials

Let $$p(z), q(z)$$and $$r(z)$$ be polynomials with complex coefficients in the complex plane. Suppose that $$|p(z)| + |q(z)| \leq |r(z)|$$ for every $$z$$. Show that there exist two complex numbers $$a,b$$ such that $$|a|^2 +|b|^2 =1$$ and $$a p(z) + bq(z) =0$$ for every $$z$$.
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# Solution: 2023-13 Range of a function on subintervals

Prove or disprove the existence of a function $$f:[0, 1] \to [0, 1]$$ with the following property:

for any interval $$(a, b) \subset [0, 1]$$ with $$a<b$$, $$f((a, b)) = [0, 1]$$.

The best solution was submitted by 조현준 (KAIST 수리과학과 22학번, +4). Congratulations!

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 오동언 (서울대학교 의과대학 19하번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), Eun U (+3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +2), 이명규 (KAIST 전산학부 20학번, +2), Anar Rzayev (KAIST 전산학부 19학번, +2). There were incorrect solutions.

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# 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 $$\frac{f(t)}{g(t)} = t^2-t+1$$.)

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Prove or disprove the existence of a function $$f:[0, 1] \to [0, 1]$$ with the following property:
for any interval $$(a, b) \subset [0, 1]$$ with $$a<b$$, $$f((a, b)) = [0, 1]$$.