Show that there exist perfect squares a, b, c such that $a^2 + b^2 = c^2$.

====== REVISED (2022-04-04) ======

I hope you noticed the day this problem appeared was April fool’s day. Show instead that there do not exist perfect squares a, b, c such that $a^2 + b^2 = c^2$, provided that a, b, c are nonzero integers.

For $k,n\geq 1$, let $v_1,\dots, v_n$ be unit vectors in $\mathbb{R}^k$. Prove that we can always choose signs $\varepsilon_1,\dots,\varepsilon_n\in \{-1, +1\}$ such that $|\sum_{i=1}^{n} \varepsilon_i v_i |\leq \sqrt{n}$.

For any positive integer $n \geq 2$, let $B_n$ be the group given by the following presentation $$B_n = < \sigma_1, \ldots, \sigma_{n-1} | \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \sigma_i \sigma_j = \sigma_j \sigma_i >$$ where the first relation is for $1 \leq i \leq n-2$ and the second relation is for $|i-j| \geq 2$. Show that there exists a total order < on $B_n$ such that for any three elements $a, b, c\in B_n$, if $a < b$ then $ca < cb$. 

There are $n$ people participating to a chess tournament and every two players play one game. There are no draws. Let $a_i$ be the number of wins of the $i$-th player and $b_i$ be the number of losses of the $i$-th player. Prove that
$$\sum_{i\in [n]} a_i^2 = \sum_{i\in [n]} b_i^2.$$

Let $n, m$ be positive integers where $m$ divides $n$. When there exists a regular $n$-gon with area 1, what is the area of the largest regular $m$-gon inscribed in the $n$-gon in terms of $n$ and $m$?

The problem on 2021-21 was written in an ambiguous way, which led the contestants to misunderstand the problem. The problem is updated to be more clear, and anyone is again welcome to submit a solution for the problem.

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. In other words, for $F =\{ A_1,A_2,\dots, A_k\},$ there exists no two sets $I, J\subseteq [k]$ with $I\cap J =\emptyset$ and $\bigcup_{i\in I}A_i = \bigcup_{j\in J} A_j$. Determine the maximum possible size of $F$.