MathDB

16

Part of 2004 Romania Team Selection Test

Problems(1)

again circles and this time some radii inequalities

Source: Romanian IMO Team Selection Test TST 2004, problem 16

5/26/2004
Three circles K1\mathcal{K}_1, K2\mathcal{K}_2, K3\mathcal{K}_3 of radii R1,R2,R3R_1,R_2,R_3 respectively, pass through the point OO and intersect two by two in A,B,CA,B,C. The point OO lies inside the triangle ABCABC.
Let A1,B1,C1A_1,B_1,C_1 be the intersection points of the lines AO,BO,COAO,BO,CO with the sides BC,CA,ABBC,CA,AB of the triangle ABCABC. Let α=OA1AA1 \alpha = \frac {OA_1}{AA_1} , β=OB1BB1 \beta= \frac {OB_1}{BB_1} and γ=OC1CC1 \gamma = \frac {OC_1}{CC_1} and let RR be the circumradius of the triangle ABCABC. Prove that αR1+βR2+γR3≥R. \alpha R_1 + \beta R_2 + \gamma R_3 \geq R.
inequalitiesgeometrycircumcirclegeometric transformationhomothetyparallelogramcomplex numbers