MathDB

2

Part of 2008 Vietnam Team Selection Test

Problems(2)

polynomials

Source: Vietnam TST 2008, Problem 2

9/5/2008
Find all values of the positive integer m m such that there exists polynomials P(x),Q(x),R(x,y) P(x),Q(x),R(x,y) with real coefficient satisfying the condition: For every real numbers a,b a,b which satisfying am−b2=0 a^m-b^2=0, we always have that P(R(a,b))=a P(R(a,b))=a and Q(R(a,b))=b Q(R(a,b))=b.
algebrapolynomialalgebra proposed
three circles

Source: Vietnam TST 2008, Problem 5

9/5/2008
Let k k be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that \frac {AL}{AD} \equal{} \frac {BM}{BE} \equal{} \frac {CN}{CF} \equal{} k. Denote (O1),(O2),(O3) (O_1),(O_2),(O_3) be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C. 1) Prove that when k \equal{} \frac{1}{2}, three circles (O1),(O2),(O3) (O_1),(O_2),(O_3) have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles. 2) Find all values of k such that three circles (O1),(O2),(O3) (O_1),(O_2),(O_3) have exactly two common points
geometrycircumcircleEulerpower of a pointradical axisgeometry proposed