MathDB

1

Part of 2011 South East Mathematical Olympiad

Problems(2)

Function problem

Source: China South East Mathematical Olympiad2011

8/18/2011
If min⁡{ax2+bx2+1∣x∈R}=3\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3, then (1) Find the range of bb; (2) for every given bb, find aa.
functioncalculusderivativealgebra proposedalgebra
Collinear problem

Source: China South East Mathematical Olympiad 2011

8/18/2011
In triangle ABCABC , AA0,BB0,CC0AA_0,BB_0,CC_0 are the angle bisectors , A0,B0,C0A_0,B_0,C_0are on sides BC,CA,AB,BC,CA,AB, draw A0A1//BB0,A0A2//CC0A_0A_1//BB_0,A_0A_2//CC_0 ,A1A_1 lies on ACAC ,A2A_2 lies on ABAB , A1A2A_1A_2 intersects BCBC at A3A_3.B3B_3 ,C3C_3 are constructed similarly.Prove that : A3,B3,C3A_3,B_3,C_3 are collinear.
geometry proposedgeometry