KURSCHAK 2003

Draw a circle $k$ with diameter $\overline{EF}$ , and let its tangent in $E$ be $e$ . Consider all possible pairs $A,B\in e$ for which $E\in \overline{AB}$ and $AE\cdot EB$ is a fixed constant. Define $(A_1,B_1)=(AF\cap k,BF\cap k)$ . Prove that the segments $\overline{A_1B_1}$ all concur in one point.

Prove that if a graph $\mathcal{G}$ on $n\ge 3$ vertices has a unique $3$ -coloring, then $\mathcal{G}$ has at least $2n-3$ edges.

(A graph is $3$ -colorable when there exists a coloring of its vertices with $3$ colors such that no two vertices of the same color are connected by an edge. The graph can be $3$ -colored uniquely if there do not exist vertices $u$ and $v$ of the graph that are painted different colors in one $3$ -coloring, yet are colored the same in another.)

Prove that the following inequality holds with the exception of finitely many positive integers $n$ :

$$\sum_{i=1}^n\sum_{j=1}^n gcd(i,j)>4n^2.$$