Korea North TST 2013 2p

 geometry problems from North Korean Team Selection Tests (TST) with aops links in the names

2013 only

2013 North Korea TST p1

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC, CA, AB$ at $A_1 , B_1 , C_1$ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2$. The line $B_1 C_1$ meets the line $BC$ at $A_3$ and the line $A_2 A_3$ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 )$. Define $B_4 , C_4$ similarly. Prove that the lines $AA_4 , BB_4 , CC_4$ are concurrent.

2013 North Korea TST p5

The incircle $\omega$ of a quadrilateral $ABCD$ touches $AB, BC, CD, DA$ at $E, F, G, H$, respectively. Choose an arbitrary point $X$ on the segment $AC$ inside $\omega$. The segments $XB, XD$ meet $\omega$ at $I, J$ respectively. Prove that $FJ, IG, AC$ are concurrent.