MathDB

2

Part of 2004 Regional Olympiad - Republic of Srpska

Problems(4)

Regional Olympiad - Republic of Srpska 2004 Grade 9 Problem 2

Source: Regional Olympiad - Republic of Srpska 2004

9/19/2018
Let ABCABC be a triangle, TT its centroid and SS its incenter. Prove that the following conditions are equivalent:
(1) line TSTS is parallel to one side of triangle ABCABC,
(2) one of the sides of triangle ABCABC is equal to the half-sum of the other two sides.
geometryCentroidparallelincenter
rs geometry

Source: RS2004

3/20/2005
Given an isosceles triangle ABCABC with base ABAB, cirumcenter OO, incenter SS and ∠C<60∘\angle C<60^\circ. The circumcircle of AOSAOS intersects ACAC at DD. Prove that SD∥BCSD\parallel BC and AS⊥ODAS\perp OD.
geometryincentercircumcirclegeometry proposed
sin ineq

Source: RS2004

3/20/2005
Let 0<x<π/20<x<\pi/2. Prove the inequality sin⁡x>4xx2+4.\sin x>\frac{4x}{x^2+4} .
inequalitiestrigonometryinequalities proposed
interesting one

Source: RS2004

3/20/2005
The positive real numbers x,y,zx,y,z satisfy x+y+z=1x+y+z=1. Show that 3xyz(1x+1y+1z+11−x+11−y+11−z)≥4+4xyz(1−x)(1−y)(1−z).\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+ \frac{4xyz}{(1-x)(1-y)(1-z)}.
inequalities proposedinequalities