Category Archives: problem

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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2022-12 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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2022-08 two sequences

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 \(n \neq 3\)?

Solutions are due May 13th (Friday), 6PM, and it will remain open if nobody solved it.

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