Let \(F\) be a family of nonempty subsets of \([n]=\{1,\dots,n\}\) such that no two disjoint subsets of \(F\) have the same union. Determine the maximum possible size of \(F\).

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Here is the best solution of problem 2021-21.

Say a natural number \(n\) is a cyclically perfect if one can arrange the numbers from 1 to \(n\) on the circle without a repeat so that the sum of any two consecutive numbers is a perfect square. Show that 32 is the smallest cyclically perfect number. Find the second smallest cyclically perfect number.

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Here is the best solution of problem 2021-20.

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.

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.