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IMO Shortlist 2011, G2

Source: IMO Shortlist 2011, G2

July 13, 2012
geometrycircumcirclealgebrapolynomialquadraticscomplex numbersIMO Shortlist

Problem Statement

Let A1A2A3A4A_1A_2A_3A_4 be a non-cyclic quadrilateral. Let O1O_1 and r1r_1 be the circumcentre and the circumradius of the triangle A2A3A4A_2A_3A_4. Define O2,O3,O4O_2,O_3,O_4 and r2,r3,r4r_2,r_3,r_4 in a similar way. Prove that 1O1A12−r12+1O2A22−r22+1O3A32−r32+1O4A42−r42=0.\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.
Proposed by Alexey Gladkich, Israel
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