VN-TST 2017 Problem 3

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.