MathDB

1

Part of 2007 Bulgaria Team Selection Test

Problems(2)

Extremal geometry problem

Source: Bulgarian TST 2007 for Balkan MO and ARO, I day Problem 1

4/9/2007
Let ABCABC is a triangle with ∠BAC=π6\angle BAC=\frac{\pi}{6} and the circumradius equal to 1. If XX is a point inside or in its boundary let m(X)=min⁡(AX,BX,CX).m(X)=\min(AX,BX,CX). Find all the angles of this triangle if max⁡(m(X))=33.\max(m(X))=\frac{\sqrt{3}}{3}.
geometrycircumcircletrigonometryinequalitiesgeometry proposed
Isosceles triangle

Source: Bulgarian TST 2007 for Balkan MO and ARO, II day Problem 1

4/9/2007
In isosceles triangle ABC(AC=BC)ABC(AC=BC) the point MM is in the segment ABAB such that AM=2MB,AM=2MB, FF is the midpoint of BCBC and HH is the orthogonal projection of MM in AF.AF. Prove that ∠BHF=∠ABC.\angle BHF=\angle ABC.
trigonometrygeometrygeometric transformationhomothetypower of a pointradical axisgeometry proposed