Proving the inradius sum up of two triangle from segment ABC is least from the inradius triangle ABC (Solution Geometry Shortlist IMO 2014)

Problem 2: Let ABC be a triangle. The points K, L, and M lie on the segments BC, CA, and AB, respectively, such that the lines AK, BL, and CM intersect in a common point. Prove that it is...

Proving that two line intersect on the circumcircle of the triangle (Solution Shortlist Geometry problem in IMO 2014)

Problem 1 The points P and Q are chosen on the side BC of an acute-angled triangle ABC so that and . The points M and N are taken on the rays AP and AQ, respectively, so that AP...

Geometry shortlist problem in IMO 2014

Geometry Problem 1 The points P and Q are chosen on the side BC of an acute-angled triangle ABC so that and . The points M and N are taken on the rays AP and AQ, respectively, so that AP...

Proving that we have to dissect R into at least (n + 1) smaller rectangles ( solution Problem 1 shortlist Problem IMO 2014 about Combinatorics )

Problem 1: Let n points be given inside a rectangle R such that no two of them lie on a line parallel to one of the sides of R. The rectangle R is to be dissected into smaller rectangles...

Solution problem 2 of shortlist problem geometry in IMO 2015

Problem 2. Let ABC be a triangle inscribed into a circle with center O. A circle with center A meets the side BC at points D and E such that D lies between B and E. Moreover, let F...

Solution problem 1 of Geometry Shortlist Problem in IMO 2015

Problem 1. Let ABC be an acute triangle with orthocenter H. Let G be the point such that the quadrilateral ABGH is a parallelogram. Let I be the point on the line GH such that AC bisects HI. Suppose...

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