52 Mathematical Olympiad

Shortlisted Problems with Solutions

Prove that the quadrilateral BXCY is cyclic. G5. Let ABC be a triangle with ∠BCA = 90 , and let C 0 be the foot of the altitude from C. Choose a point X in the interior of the segment CC 0, and let K,L be the points on the segments AX,BX for which BK = BC and AL = AC respectively. Denote by M the intersection of AL and BK. Show that MK = ML. G6.

  Cyclic

Shortlisted Problems with Solutions

Shortlisted problems 3 Problems Algebra A1. Let nbe a positive integer and let a 1,...,an´1 be arbitrary real numbers. Define the sequences u 0,...,un and v 0,...,vn inductively by u 0 “ u 1 “ v 0 “ v 1 “ 1, and uk`1 “ uk `akuk´1, vk`1 “ vk `an´kvk´1 for k“ 1,...,n´1. Prove that un “ vn. (France) A2. Prove that in any set of 2000 distinct real numbers there exist two ...

  With, Solutions, Problem, Shortlisted, Shortlisted problems with solutions

IMO2018 Shortlisted Problems with Solutions

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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.

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