# 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!

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# Notice on POW 2021-17, 2021-18

POW 2021-17 and 2021-18 are still open. For each problem, anyone who first submits a correct solution will get the full credit.

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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!

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# 2021-17 distributing points on the sphere

On the sphere of radius r, how many points one can put if any two points must be distance at least 1 apart? How fast does this number grow as a function in r?

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