Team selection tests for JBMO 2018

Denote $X$ as the other intersection of $(IJL)$ and $(\omega )$. We shall prove that $X$ is the intersection of $EL$ and $FJ$.

By the cyclic quadrilateral, we have $$\angle LXI=\angle LJI=\angle LJB=\angle LCB=\angle LCA=180^{\circ }-\angle EXI.$$ Hence $L$, $X$, $E$ are collinear, which means $X$ belongs to $EL$.

Similarly, $JF$ passes through $X$. These finish our proof.