MathDB

1

Part of 2006 South East Mathematical Olympiad

Problems(2)

problem about function

Source: China south east mathematical Olympiad 2006 problem1

7/4/2013
Suppose a>b>0a>b>0, f(x)=2(a+b)x+2ab4x+a+bf(x)=\dfrac{2(a+b)x+2ab}{4x+a+b}. Show that there exists an unique positive number xx, such that f(x)=(a13+b132)3f(x)=\left(\dfrac{a^{\frac{1}{3}}+b^{\frac{1}{3}}}{2} \right)^3.
functionalgebra unsolvedalgebra
incircle

Source: China south east mathematical olympiad 2006 day2 problem 5

7/4/2013
In △ABC\triangle ABC, ∠A=60∘\angle A=60^\circ. ⊙I\odot I is the incircle of △ABC\triangle ABC. ⊙I\odot I is tangent to sides ABAB, ACAC at DD, EE, respectively. Line DEDE intersects line BIBI and CICI at FF, GG respectively. Prove that FG=BC2FG=\frac{BC}{2}.
geometrytrigonometrysymmetrygeometry unsolved