MathDB

3

Part of 2013 CentroAmerican

Problems(2)

ABCD is a Convex Quadrilateral

Source: CentroAmerican 2013 Problem 3

8/24/2013
Let ABCDABCD be a convex quadrilateral and let MM be the midpoint of side ABAB. The circle passing through DD and tangent to ABAB at AA intersects the segment DMDM at EE. The circle passing through CC and tangent to ABAB at BB intersects the segment CMCM at FF. Suppose that the lines AFAF and BEBE intersect at a point which belongs to the perpendicular bisector of side ABAB. Prove that AA, EE, and CC are collinear if and only if BB, FF, and DD are collinear.
geometryperpendicular bisectorgeometry unsolved
p(x)-q(x)=1

Source: CentroAmerican 2013 Problem 6

8/24/2013
Determine all pairs of non-constant polynomials p(x)p(x) and q(x)q(x), each with leading coefficient 11, degree nn, and nn roots which are non-negative integers, that satisfy p(x)āˆ’q(x)=1p(x)-q(x)=1.
algebrapolynomialalgebra unsolved