RO-TST 2004 Problem 16

Three circles $\mathcal{K}_1$ , $\mathcal{K}_2$ , $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$ . The point $O$ lies inside the triangle $ABC$ .

Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$ . Let $\alpha = \frac {OA_1}{AA_1}$ , $\beta= \frac {OB_1}{BB_1}$ and $\gamma = \frac {OC_1}{CC_1}$ and let $R$ be the circumradius of the triangle $ABC$ . Prove that $$\alpha R_1 + \beta R_2 + \gamma R_3 \geq R.$$