MathDB

1

Part of 2007 China National Olympiad

Problems(2)

China Mathematics Olympiads (National Round) 2007 Problem 1

Source:

11/28/2010
Given complex numbers a,b,ca, b, c, let ∣a+b∣=m,∣a−b∣=n|a+b|=m, |a-b|=n. If mn≠0mn \neq 0, Show that max⁡{∣ac+b∣,∣a+bc∣}≥mnm2+n2\max \{|ac+b|,|a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}
complex numbersinequalities proposedinequalities
China Mathematics Olympiads (National Round) 2007 Problem 4

Source:

11/28/2010
Let O,IO, I be the circumcenter and incenter of triangle ABCABC. The incircle of △ABC\triangle ABC touches BC,CA,ABBC, CA, AB at points D,E,FD, E, F repsectively. FDFD meets CACA at PP, EDED meets ABAB at QQ. MM and NN are midpoints of PEPE and QFQF respectively. Show that OI⊥MNOI \perp MN.
geometrycircumcircleincentertrigonometrypower of a pointradical axisgeometry unsolved