Team selection tests for IMO 2018

The solution is based on a simple fact, that if you add the arithmetic mean of the sequence to the sequence, then the arithmetic mean of the sequence will not change, since $$\frac{a_1+a_2+\cdots +a_n}{n}=\frac{a_1+a_2+\cdots +a_n+\frac{a_1+a_2+\cdots +a_n}{n}}{n+1}$$ Denote by $\mu (A)$ the arithmetic mean of the elements of $A$. Denote $$P=\left\{A\subset \{1,2,\dots ,n\}:\mu (A)\in {\mathbb{Z}}_{+},\mu (A)\in A\right\}$$ and $$Q=\left\{A\subset \{1,2,\dots ,n\}:\mu (A)\in {\mathbb{Z}}_{+},\mu (A)\notin A\right\}.$$ We need to prove that $|P|+|Q|$ is even. For that we will prove $|P|=|Q|$. Really, take any set from $P$ and remove its arithmetic mean. We will get an element from $Q$. Take any set from $Q$ and add its arithmetic mean. We will get an element from $P$. Since arithmetic mean of the set is defined uniquely, so we have bijection between $P$ and $Q$.