MathDB

2

Part of 2003 Vietnam National Olympiad

Problems(2)

real roots of two plynomials

Source: 41-st Vietnamese Mathematical Olympiad 2003

3/10/2007
Define p(x)=4x3−2x2−15x+9,q(x)=12x3+6x2−7x+1p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1. Show that each polynomial has just three distinct real roots. Let AA be the largest root of p(x)p(x) and BB the largest root of q(x)q(x). Show that A2+3B2=4A^{2}+3 B^{2}= 4.
algebrapolynomialalgebra proposed
Straight line

Source: 41-st Vietnamese Mathematical Olympiad 2003

8/28/2004
The circles C1 C_{1} and C2 C_{2} touch externally at M M and the radius of C2 C_{2} is larger than that of C1 C_{1}. A A is any point on C2 C_{2} which does not lie on the line joining the centers of the circles. B B and C C are points on C1 C_{1} such that AB AB and AC AC are tangent to C1 C_{1}. The lines BM BM, CM CM intersect C2 C_{2} again at E E, F F respectively. D D is the intersection of the tangent at A A and the line EF EF. Show that the locus of D D as A A varies is a straight line.
geometrygeometric transformationhomothetyinvarianttrigonometryblogsratio