There will be no POW this week due to 추석 (thanksgiving) break. POW will resume next week.
Monthly Archives: September 2023
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!
Here is the best solution of problem 2023-15.
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).
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 \( x_n = 0 \).
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!
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).
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 \(z\).
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!
Here is the best solution of problem 2023-13.
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.
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 \(\frac{f(t)}{g(t)} = t^2-t+1\).)
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 \( a<b \), \( f((a, b)) = [0, 1] \).