# 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
$\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.$

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# 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!

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.

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# Solution: 2023-05 Shuffle, multiply, and add

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!

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# Notice: Mid-term break

POW will resume on Apr. 28.

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# 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.

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# 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.

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