MathDB

2

Part of 2015 Moldova Team Selection Test

Problems(3)

Inequality with product equal to 1

Source: Moldova TST Problem 6

4/1/2015
Let a,b,ca,b,c be positive real numbers such that abc=1abc=1. Prove the following inequality: \\a3+b3+c3+aba2+b2+bcb2+c2+cac2+a2≥92a^3+b^3+c^3+\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2} \geq \frac{9}{2}.
inequalities
Collinearity of points on angle bisectors

Source: Moldova TST Problem 2

3/31/2015
Consider a triangle △ABC\triangle ABC, let the incircle centered at II touch the sides BC,CA,ABBC,CA,AB at points D,E,FD,E,F respectively. Let the angle bisector of ∠BIC\angle BIC meet BCBC at MM, and the angle bisector of ∠EDF\angle EDF meet EFEF at NN. Prove that A,M,NA,M,N are collinear.
geometryincenterangle bisector
Nice trigonometry problem

Source:

3/31/2015
Prove the equality:\\ tan⁡(3π7)−4sin⁡(π7)=7\tan (\frac{3\pi}{7})-4\sin (\frac{\pi}{7})= \sqrt{7} .
trigonometryalgebra