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
\[ \left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx. \]
Let \(a(n)\) be the number of unordered factorizations of \(n\) into divisors larger than \(1\). Prove that \(\sum_{n=2}^{\infty} \frac{a(n)}{n^2} = 1\).
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 \[ \sum_{i=1}^n \phi(n,i).\]
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 \( x \in \mathbb{R} \).
Let \(S(n,k)\) be the Stirling number of the second kind that is the number of ways to partition a set of \(n\) objects into \(k\) non-empty subsets. Prove the following equality \[ \det\left( \begin{matrix} S(m+1,1) & S(m+1,2) & \cdots & S(m+1,n) \\
S(m+2,1) & S(m+2,2) & \cdots & S(m+2,n) \\
\cdots & \cdots & \cdots & \cdots \\
S(m+n,1) & S(m+n,2) & \cdots & S(m+n,n) \end{matrix} \right) = (n!)^m \]
For a positive integer \(n\), let \(B\) and \(C\) be real-valued \(n\) by \(n\) matrices and \(O\) be the \(n\) by \(n\) zero matrix. Assume further that \(B\) is invertible and \(C\) is symmetric. Define \[A := \begin{pmatrix} O & B \\ B^T & C \end{pmatrix}.\] What is the possible number of positive eigenvalues for \(A\)?
Prove for any \( x \geq 1 \) that
\[
\left( \sum_{n=0}^{\infty} (n+x)^{-2} \right)^2 \geq 2 \sum_{n=0}^{\infty} (n+x)^{-3}.
\]
Consider the power set \(P([n])\) consisting of \(2^n\) subsets of \([n]=\{1,\dots,n\}\).
Find the smallest \(k\) such that the following holds: there exists a partition \(Q_1,\dots, Q_k\) of \(P([n])\) so that there do not exist two distinct sets \(A,B\in P([n])\) and \(i\in [k]\) with \(A,B,A\cup B, A\cap B \in Q_i\).
Does there exists a finitely generated group which contains torsion elements of order p for all prime numbers p?
Solutions for POW 2022-11 are due July 4th (Saturday), 12PM, and it will remain open if nobody solved it.
Prove or disprove the following:
For any positive integer \( n \), there exists a polynomial \( P_n \) of degree \( n^2 \) such that
(1) all coefficients of \( P_n \) are integers with absolute value at most \( n^2 \), and
(2) \( 1 \) is a root of \( P_n =0 \) with multiplicity at least \( n \).