RU-SPB 2020 Problem 115

The altitudes $BB_1$ and $CC_1$ of the acute triangle $\triangle ABC$ intersect at $H$ . The circle centered at $O_b$ passes through points $A,C_1$ , and the midpoint of $BH$ . The circle centered at $O_c$ passes through $A,B_1$ and the midpoint of $CH$ . Prove that $B_1 O_b +C_1O_c > \frac{BC}{4}$