VN-TST 2017

For each positive integer $n$ , set $x_n=\binom{2n}{n}$ . a. Prove that if $\frac{2017^k}{2}<n<2017^k$ for some positive integer $k$ then $2017$ divides $x_n$ . b. Find all positive integer $h>1$ such that there exists positive integers $N,T$ such that $(x_n)_{n>N}$ is periodic mod $h$ with period $T$ .

Triangle $ABC$ with incircle $(I)$ touches the sides $AB, BC, AC$ at $F, D, E$ , res. $I_b, I_c$ are $B$ - and $C-$ excenters of $ABC$ . $P, Q$ are midpoints of $I_bE, I_cF$ . $(PAC)\cap AB=\{ A, R\}$ , $(QAB)\cap AC=\{ A,S\}$ . a. Prove that $PR, QS, AI$ are concurrent. b. $DE, DF$ cut $I_bI_c$ at $K, J$ , res. $EJ\cap FK=\{ M\}$ . $PE, QF$ cut $(PAC), (QAB)$ at $X, Y$ res. Prove that $BY, CX, AM$ are concurrent.