# 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 $\sum_{i=1}^n \phi(n,i).$

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# 2022-16 Identity for continuous functions

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 $$x \in \mathbb{R}$$.

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# 2022-15 A determinant of Stirling numbers of second kind

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 $\det\left( \begin{matrix} S(m+1,1) & S(m+1,2) & \cdots & S(m+1,n) \\ S(m+2,1) & S(m+2,2) & \cdots & S(m+2,n) \\ \cdots & \cdots & \cdots & \cdots \\ S(m+n,1) & S(m+n,2) & \cdots & S(m+n,n) \end{matrix} \right) = (n!)^m$

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# 2022-14 The number of eigenvalues of a symmetric matrix

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 $$A$$?

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# 2022-13 Inequality involving sums with different powers

Prove for any $$x \geq 1$$ that

$\left( \sum_{n=0}^{\infty} (n+x)^{-2} \right)^2 \geq 2 \sum_{n=0}^{\infty} (n+x)^{-3}.$

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# 2022-12 A partition of the power set of a set

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 $$A,B,A\cup B, A\cap B \in Q_i$$.

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# 2022-11 groups with torsions

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.

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# 2022-10 Polynomial with root 1

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 $$n$$.

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# 2022-09 A chaotic election

Let $$A_1,\dots, A_k$$ be presidential candidates in a country with $$n \geq 1$$ voters with $$k\geq 2$$. Candidates themselves are not voters. Each voter has her/his own preference on those $$k$$ candidates.

Find maximum $$m$$ such that the following scenario is possible where $$A_{k+1}$$ indicates the candidate $$A_1$$: for each $$i\in [k]$$, there are at least $$m$$ voters who prefers $$A_i$$ to $$A_{i+1}$$.

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For positive integers $$n \geq 2$$, let $$a_n = \lceil n/\pi \rceil$$ and let $$b_n = \lceil \csc (\pi/n) \rceil$$. Is $$a_n = b_n$$ for all $$n \neq 3$$?