MathDB

2

Part of 2009 Korea National Olympiad

Problems(2)

Easy 3 variables inequality

Source: Korea National Olympiad 2009 Problem 2

9/9/2012
Let a,b,c a,b,c be positive real numbers. Prove that a3c(a2+bc)+b3a(b2+ca)+c3b(c2+ab)≥32. \frac{a^3}{c(a^2 + bc)} + \frac{b^3}{a(b^2 + ca)} + \frac{c^3}{b(c^2 +ab )} \ge \frac{3}{2} .
inequalitiesinequalities proposed
Convex quadrilateral

Source: Korea National Olympiad 2009 Problem 6

9/9/2012
Let ABCABC be a triangle and P,Q(≠A,B,C) P, Q ( \ne A, B, C ) are the points lying on segments BC,CA BC , CA . Let I,J,K I, J, K be the incenters of triangle ABP,APQ,CPQ ABP, APQ, CPQ . Prove that PIJK PIJK is a convex quadrilateral.
geometryincentergeometry proposed