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2

Part of 2003 Turkey MO (2nd round)

Problems(2)

Geometric inequality.

Source: Turkey MO 2004.

2/27/2005
Let ABCDABCD be a convex quadrilateral and K,L,M,NK,L,M,N be points on [AB],[BC],[CD],[DA][AB],[BC],[CD],[DA], respectively. Show that, s13+s23+s33+s43≤2s3 \sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s} where s1=Area(AKN)s_1=\text{Area}(AKN), s2=Area(BKL)s_2=\text{Area}(BKL), s3=Area(CLM)s_3=\text{Area}(CLM), s4=Area(DMN)s_4=\text{Area}(DMN) and s=Area(ABCD)s=\text{Area}(ABCD).
geometryinequalitiesparallelograminequalities solvedGeometric Inequalities
I incenter show that ...

Source: Turkey NMO 2003 Problem 5

2/24/2009
A circle which is tangent to the sides [AB] [AB] and [BC] [BC] of △ABC \triangle ABC is also tangent to its circumcircle at the point T T. If I I is the incenter of △ABC \triangle ABC , show that \widehat{ATI}\equal{}\widehat{CTI}
geometryincentercircumcirclesearchgeometry unsolved