BD-BDMO 2023

Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$ . The line tangent to $\omega$ at $N$ intersects $BC$ at $T$ , $NS$ intersects $\omega$ at $K$ . Assume that $\angle NTC = \angle KSB$ . Prove that $CK\parallel AN \parallel TS$ .

We are given $n$ intervals $[l_1,r_1],[l_2,r_2],[l_3,r_3],\dots, [l_n,r_n]$ in the number line. We can divide the intervals into two sets such that no two intervals in the same set have overlaps. Prove that there are at most $n-1$ pairs of overlapping intervals.