SHARYGIN 2010 Problem 15

Let $AA_1, BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC$, $AA_1$ meets $B_1C_1$ in a point $K$. The circumcircles of the triangles $A_1KC_1$ and $A_1KB_1$ intersect the lines $AB$ and $AC$ for the second time at points $N$ and $L$ respectively. Prove that

a) the sum of diameters of these two circles is equal to $BC$; b) $A_1N/BB_1 + A_1L/CC_1 = 1$.