MathDB

3

Part of 2007 CentroAmerican

Problems(2)

Quadratic polynomials and a set of integers

Source: Central American Olympiad 2007, Problem 3

6/12/2007
Let SS be a finite set of integers. Suppose that for every two different elements of SS, pp and qq, there exist not necessarily distinct integers a≠0a \neq 0, bb, cc belonging to SS, such that pp and qq are the roots of the polynomial ax2+bx+cax^{2}+bx+c. Determine the maximum number of elements that SS can have.
quadraticsalgebrapolynomialalgebra proposed
Nice problem from a friend: prove G is the centroid of ABP

Source: Central American Olympiad 2007, Problem 6

6/12/2007
Consider a circle SS, and a point PP outside it. The tangent lines from PP meet SS at AA and BB, respectively. Let MM be the midpoint of ABAB. The perpendicular bisector of AMAM meets SS in a point CC lying inside the triangle ABPABP. ACAC intersects PMPM at GG, and PMPM meets SS in a point DD lying outside the triangle ABPABP. If BDBD is parallel to ACAC, show that GG is the centroid of the triangle ABPABP. Arnoldo Aguilar (El Salvador)
geometryparallelogramincenterperpendicular bisectorgeometry proposed