# 2022-19 Inequality for twice differentiable functions

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
$\left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx.$

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# Solution: 2022-18 A sum of the number of factorizations

Let $$a(n)$$ be the number of unordered factorizations of $$n$$ into divisors larger than $$1$$. Prove that $$\sum_{n=2}^{\infty} \frac{a(n)}{n^2} = 1$$.

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

Other solutions were submitted by 기영인 (KAIST 22학번, +3), Kawano Ren (Kaisei Senior High School, +3), Sakae Fujimoto (Osaka Prefectural Kitano High School, Freshmen, +3), 최백규 (KAIST 생명과학과 20학번, +3).

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# Solution: 2022-17 The smallest number of subsets

Let $$n, i$$ be integers such that $$1 \leq i \leq n$$. Each subset of $$\{ 1, 2, \ldots, n \}$$ with $$i$$ elements has the smallest number. We define $$\phi(n,i)$$ to be the sum of these smallest numbers. Compute $\sum_{i=1}^n \phi(n,i).$

The best solution was submitted by 김유준 (KAIST 수리과학과 20학번, +4). Congratulations!

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 기영인 (KAIST 22학번, +3), 이준환 (한국외국어대학교 통계학과 19학번, +3), 오준혁 (KAIST 수리과학과 20학번, +3), 신준범 (컬럼비아 대학교 20학번, +3), 이한스 (KAIST 수리과학과 20학번, +3), Kawano Ren (Kaisei Senior High School, +3), Sakae Fujimoto (Osaka Prefectural Kitano High School, Freshmen, +3).

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# Notice: Mid-term break

POW will resume on Oct. 28.

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# 2022-18 A sum of the number of factorizations

Let $$a(n)$$ be the number of unordered factorizations of $$n$$ into divisors larger than $$1$$. Prove that $$\sum_{n=2}^{\infty} \frac{a(n)}{n^2} = 1$$.

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For a positive integer $$n$$, find all continuous functions $$f: \mathbb{R} \to \mathbb{R}$$ such that
$\sum_{k=0}^n \binom{n}{k} f(x^{2^k}) = 0$
for all $$x \in \mathbb{R}$$.