MathDB

1

Part of 2000 China Team Selection Test

Problems(2)

China TST 2000 circumcircle of triangle ADE

Source: China TST 2000, problem 1

5/22/2005
Let ABCABC be a triangle such that AB=ACAB = AC. Let D,ED,E be points on AB,ACAB,AC respectively such that DE=ACDE = AC. Let DEDE meet the circumcircle of triangle ABCABC at point TT. Let PP be a point on ATAT. Prove that PD+PE=ATPD + PE = AT if and only if PP lies on the circumcircle of triangle ADEADE.
geometrycircumcircleconicsellipseangle bisectorgeometry unsolved
Coefficients of Gamma function

Source: China TST 2000, problem 4

5/22/2005
Let FF be the set of all polynomials Γ\Gamma such that all the coefficients of Γ(x)\Gamma (x) are integers and Γ(x)=1\Gamma (x) = 1 has integer roots. Given a positive intger kk, find the smallest integer m(k)>1m(k) > 1 such that there exist Γ∈F\Gamma \in F for which Γ(x)=m(k)\Gamma (x) = m(k) has exactly kk distinct integer roots.
functionalgebrapolynomialalgebra unsolved