MathDB

8

Part of 2007 Hanoi Open Mathematics Competitions

Problems(2)

HOMC (Vietnam)

Source:

1/28/2016
Let a; b; c be positive integers. Prove that (b+c−a)2(b+c)2+a2+(c+a−b)2(c+a)2+b2+(a+b−c)2(a+b)2+c2≥35 \frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac{3}{5}
inequalities
locus inside an equilateral triangle (2007 HOMC Senior Q8)

Source:

7/18/2019
Let ABCABC be an equilateral triangle. For a point MM inside △ABC\vartriangle ABC, let D,E,FD,E,F be the feet of the perpendiculars from MM onto BC,CA,ABBC,CA,AB, respectively. Find the locus of all such points MM for which ∠FDE\angle FDE is a right angle.
geometryLocusEquilateral