MathDB

5

Part of 2009 Irish Math Olympiad

Problems(2)

Three Variables Inequality

Source:

9/24/2010
Hello.
Suppose aa, bb, cc are real numbers such that a+b+c=0a+b+c = 0 and a2+b2+c2=1a^{2}+b^{2}+c^{2} = 1. Prove that a2b2c2≤154a^{2}b^{2}c^{2}\leq \frac{1}{54} and determine the cases of equality.
inequalitiesinequalities open
Angle Bisectors in Triangle

Source: 2009 IrMO Paper 2 Problem 5

1/30/2018
In the triangle ABCABC we have ∣AB∣<∣AC∣|AB|<|AC|. The bisectors of the angles at BB and CC meet ACAC and ABAB at DD and EE respectively. BDBD and CECE intersect at the incenter II of △ABC\triangle ABC.
Prove that ∠BAC=60∘\angle BAC=60^\circ if and only if ∣IE∣=∣ID∣|IE|=|ID|
geometryangle bisector