Solution: 2023-07 An oscillatory integral

Suppose that \( f: [a, b] \to \mathbb{R} \) is a smooth, convex function, and there exists a constant \( t>0 \) such that \( f'(x) \geq t \) for all \( x \in (a, b) \). Prove that
\[
\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.
\]

The best solution was submitted by Anar Rzayev (KAIST 전산학부 19학번, +4). Congratulations!

Here is the best solution of problem 2023-07.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 오현섭 (KAIST 수리과학과 박사과정 21학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 최예준 (서울과학기술대학교 행정학과 21학번, +3), Matthew Seok (+3), James Hamilton Clerk (+3).

2023-07 An oscillatory integral

Suppose that \( f: [a, b] \to \mathbb{R} \) is a smooth, convex function, and there exists a constant \( t>0 \) such that \( f'(x) \geq t \) for all \( x \in (a, b) \). Prove that
\[
\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.
\]

Solution: 2023-06 Golden ratio and a functionSolution:

Let \(\phi = \frac{1+\sqrt{5}}{2}\). Let \(f(1)=1\) and for \(n\geq 1\), let
\[ f(n+1) = \left\{\begin{array}{ll}
f(n)+2 & \text{ if } f(f(n)-n+1)=n \\
f(n)+1 & \text{ otherwise}.
\end{array}\right.\]
Prove that \(f(n) = \lfloor \phi n \rfloor\), and determine when \(f(f(n)-n+1)\neq n\) holds.

The best solution was submitted by 박기윤 (KAIST 새내기과정학부 23학번, +4). Congratulations!

Here is the best solution of problem 2023-06.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 이동하 (KAIST 새내기과정학부 23학번, +3), 최예준 (서울과학기술대학교 행정학과 21학번, +3), Matthew Seok (+2). Late solutions are not graded.

Notice on POW 2023-05

There were no correct solution submitted by the due (Friday 3pm). Since we received a correct solution a few hours after the due, we decided to extend the due by Apr. 14, 3pm. Any solution submitted by that due will be considered for the full credit.

2023-06 Golden ratio and a function

Let \(\phi = \frac{1+\sqrt{5}}{2}\). Let \(f(1)=1\) and for \(n\geq 1\), let
\[ f(n+1) = \left\{\begin{array}{ll}
f(n)+2 & \text{ if } f(f(n)-n+1)=n \\
f(n)+1 & \text{ otherwise}.
\end{array}\right.\]
Prove that \(f(n) = \lfloor \phi n \rfloor\), and determine when \(f(f(n)-n+1)\neq n\) holds.

Solution: 2023-04 A perfect square

Find all integers \( n \) such that \( n^4 + n^3 + n^2 + n + 1 \) is a perfect square.

The best solution was submitted by 채지석 (KAIST 수리과학과 석박사통학과정 21학번, +4). Congratulations!

Here is the best solution of problem 2023-04.

Other solutions were submitted by 기영인 (KAIST 수리과학과 22학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 노희윤 (KAIST 수리과학과 19학번, +3), 문강연 (KAIST 수리과학과 22학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 박지환 (연세대학교 수학과 22학번, +3), 백민수 (원주중학교 교사, +3), 이종서 (KAIST 전산학부 19학번, +3), Matthew Seok, 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), 이동하 (KAIST 새내기과정학부 23학번, +3).