Monthly Archives: September 2022

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

The best solution was submitted by 기영인 (KAIST 22학번, +4). Congratulations!

Here is the best solution of problem 2022-15.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김찬우 (연세대학교 수학과, +3). Late solutions were not graded.

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Solution: 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 \(A\)?

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

Here is the best solution of problem 2022-14.

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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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Solution: 2022-13 Inequality involving sums with different powers

Prove for any \( x \geq 1 \) that

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

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

Here is the best solution of problem 2022-13.

Another solution was submitted by 김찬우 (연세대학교 수학과, +3).

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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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