Monthly Archives: October 2021

2021-19 The answer is zero

Suppose that \( a_1 + a_2 + \dots + a_n =0 \) for real numbers \( a_1, a_2, \dots, a_n \) and \( n \geq 2\). Set \( a_{n+i}=a_i \) for \( i=1, 2, \dots \). Prove that
\[
\sum_{i=1}^n \frac{1}{a_i (a_i+a_{i+1}) (a_i+a_{i+1}+a_{i+2}) \dots (a_i+a_{i+1}+\dots+a_{i+n-2})} =0
\]
if the denominators are nonzero.

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Solution: 2021-18 Independent sets in a tree

Let \(T\) be a tree (an acyclic connected graph) on the vertex set \([n]=\{1,\dots, n\}\).
Let \(A\) be the adjacency matrix of \(T\), i.e., the \(n\times n\) matrix with \(A_{ij} = 1\) if \(i\) and \(j\) are adjacent in \(T\) and \(A_{ij}=0\) otherwise. Prove that the number of nonnegative eigenvalues of \(A\) equals to the size of the largest independent set of \(T\). Here, an independent set is a set of vertices where no two vertices in the set are adjacent.

The best solution was submitted by 전해구 (기계공학과 졸업생, +4). Congratulations!

Here is the best solution of problem 2021-18.

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2021-18 Independent sets in a tree

Let \(T\) be a tree (an acyclic connected graph) on the vertex set \([n]=\{1,\dots, n\}\).
Let \(A\) be the adjacency matrix of \(T\), i.e., the \(n\times n\) matrix with \(A_{ij} = 1\) if \(i\) and \(j\) are adjacent in \(T\) and \(A_{ij}=0\) otherwise. Prove that the number of nonnegative eigenvalues of \(A\) equals to the size of the largest independent set of \(T\). Here, an independent set is a set of vertices where no two vertices in the set are adjacent.

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Solution: 2021-16 Optimal constant

For a given positive integer \( n \) and a real number \( a \), find the maximum constant \( b \) such that
\[
x_1^n + x_2^n + \dots + x_n^n + a x_1 x_2 \dots x_n \geq b (x_1 + x_2 + \dots + x_n)^n
\]
for any non-negative \( x_1, x_2, \dots, x_n \).

The best solution was submitted by 전해구 (기계공학과 졸업생, +4). Congratulations!

Here is the best solution of problem 2021-16.

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