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Shortlisted Problems with Solutions

Denote by P the foot of the perpendicular from the center of ω to ℓ. The side-lines BC,CA,AB intersect ℓ at the points X,Y,Z different from P. Prove that the circumcircles of the triangles AXP,BYP and CZP have a common point different from P or are mutually tangent at P.

52 Mathematical Olympiad

Suppose also that the circumcenter O of the triangle ABC lies on the shorter arc B′C′ of ω. Prove that the circumcircle of ABC and ω meet at two points. G2 G2 Let A 1A 2A 3A 4 be a non-cyclic quadrilateral. Let O 1 and r 1 be the circumcenter and the circumradius of the triangle A 2A 3A 4. Define O 2, O 3, O 4 and r 2, r 3, r 4 in a ...

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