Let \(\varphi(x)\) be the Euler’s totient function. Let \(S = \{a_1,\dots, a_n\}\) be a set of positive integers such that for any \(a_i\), all of its positive divisors are also in \(S\). Let \(A\) be the matrix with entries \(A_{i,j} = gcd(a_i,a_j)\) being the greatest common divisors of \(a_i\) and \(a_j\). Prove that \(\det(A) = \prod_{i=1}^{n} \varphi(a_i)\).
Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function satisfying \( f(0) = 0 \) and \( 0 \leq f'(x) \leq 1 \). Prove that
\[ \left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx. \]
The best solution was submitted by 기영인 (KAIST 22학번, +4). Congratulations!
Here is the best solution of problem 2022-19.
Other solutions were submitted by 여인영 (KAIST 물리학과 20학번, +3), Kawano Ren (Kaisei Senior High School, +3), 최예준 (서울과기대 행정학과 21학번, +3), 김준성 (KAIST 물리학과 박사과정, +3).