MathDB

2

Part of 2004 Vietnam National Olympiad

Problems(2)

Vietnam NMO 2004_2

Source:

10/26/2008
In a triangle ABC ABC, the bisector of ∠ACB \angle ACB cuts the side AB AB at D D. An arbitrary circle (O) (O) passing through C C and D D meets the lines BC BC and AC AC at M M and N N (different from C C), respectively. (a) Prove that there is a circle (S) (S) touching DM DM at M M and DN DN at N N. (b) If circle (S) (S) intersects the lines BC BC and CA CA again at P P and Q Q respectively, prove that the lengths of the segments MP MP and NQ NQ are constant as (O) (O) varies.
geometryangle bisectorgeometry proposed
Very interesting

Source: Old and New Inequalities, problem 105; Vietnam MO 2004 Group A, created by Namdung

8/17/2005
Let xx, yy, zz be positive reals satisfying (x+y+z)3=32xyz\left(x+y+z\right)^{3}=32xyz Find the minimum and the maximum of P=x4+y4+z4(x+y+z)4P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}
inequalitiesalgebrapolynomialfunctionlimitquadraticsreal analysis