# Solution: 2021-21 Different unions

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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# Solution: 2021-20 A circle of perfect squares

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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# 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.

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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 $$x_1, x_2, \dots, x_n$$.