MathDB

3

Part of 2004 Vietnam Team Selection Test

Problems(2)

there are two circles intersecting each at two points

Source: VietNam TST 2004, problem 3

5/9/2004
In the plane, there are two circles Γ1,Γ2\Gamma_1, \Gamma_2 intersecting each other at two points AA and BB. Tangents of Γ1\Gamma_1 at AA and BB meet each other at KK. Let us consider an arbitrary point MM (which is different of AA and BB) on Γ1\Gamma_1. The line MAMA meets Γ2\Gamma_2 again at PP. The line MKMK meets Γ1\Gamma_1 again at CC. The line CACA meets Γ2\Gamma_2 again at QQ. Show that the midpoint of PQPQ lies on the line MCMC and the line PQPQ passes through a fixed point when MM moves on Γ1\Gamma_1. [Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .]
geometrygeometric transformationrotationhomothetyfunctionratiogeometry solved
greatest element from S

Source: VietNam TST 2004, problem 6

5/9/2004
Let SS be the set of positive integers in which the greatest and smallest elements are relatively prime. For natural nn, let SnS_n denote the set of natural numbers which can be represented as sum of at most nn elements (not necessarily different) from SS. Let aa be greatest element from SS. Prove that there are positive integer kk and integers bb such that ∣Sn∣=a⋅n+b|S_n| = a \cdot n + b for all n>k n > k .
number theory solvednumber theory