KURSCHAK 2019

In an acute triangle $\bigtriangleup ABC$ , $AB<AC<BC$ , and $A_1,B_1,C_1$ are the projections of $A,B,C$ to the corresponding sides. Let the reflection of $B_1$ wrt $CC_1$ be $Q$ , and the reflection of $C_1$ wrt $BB_1$ be $P$ . Prove that the circumcirle of $A_1PQ$ passes through the midpoint of $BC$ .

Find all family $\mathcal{F}$ of subsets of $[n]$ such that for any nonempty subset $X\subseteq [n]$ , exactly half of the elements $A\in \mathcal{F}$ satisfies that $|A\cap X|$ is even.

Is it true that if $H$ and $A$ are bounded subsets of $\mathbb{R}$ , then there exists at most one set $B$ such that $a+b(a\in A,b\in B)$ are pairwise distinct and $H=A+B$ .