I usaully dislike these type of problems, but this one is quite nice! Let the midpoint of $BH, CH$ be $N,M$ ,and the second intersection of $CC_1$ and $O_c$ be $X$, and $M_C$ the midpoint of $MX$. Now $CH\times CC_1 =CB_1 \times CA_1 =CM \times CX$ , so we get $CX=2CC_1$, which implies $OC_1 \ge C_1 M_C =\frac{CM}{2}$ Thus, we have $O_b B_1 \ge \frac{BN}{2}$ in the same way, so $O_b B_1 +O_c C_1 \ge \frac{BH+CH}{4} >\frac{BC}{4}$