MathDB

2

Part of 2010 Korea National Olympiad

Problems(2)

3 variables with condition ab+bc+ca=1

Source: Korea National Olympiad 2010 Problem 2

9/9/2012
Let a,b,c a, b, c be positive real numbers such that ab+bc+ca=1 ab+bc+ca=1 . Prove that a2+b2+1c2+b2+c2+1a2+c2+a2+1b2≥33 \sqrt{ a^2 + b^2 + \frac{1}{c^2}} + \sqrt{ b^2 + c^2 + \frac{1}{a^2}} + \sqrt{ c^2 + a^2 + \frac{1}{b^2}} \ge \sqrt{33}
inequalitiesinequalities proposed
Collinear points with orthocenters

Source: Korea National Olympiad 2010 Problem 6

9/9/2012
Let ABCD ABCD be a cyclic convex quadrilateral. Let E E be the intersection of lines AB,CD AB, CD . P P is the intersection of line passing B B and perpendicular to AC AC , and line passing C C and perpendicular to BD BD. Q Q is the intersection of line passing D D and perpendicular to AC AC , and line passing A A and perpendicular to BD BD . Prove that three points E,P,Q E, P, Q are collinear.
geometrygeometric transformationhomothetycircumcircleratioprojective geometrygeometry proposed