MathDB

1

Part of 2005 Irish Math Olympiad

Problems(2)

easy

Source: Ireland 2005

7/6/2009
Show that 20052005 2005^{2005} is a sum of two perfect squares, but not a sum of two perfect cubes.
modular arithmeticnumber theory proposednumber theory
inequality

Source: Ireland 2005

7/6/2009
Let X X be a point on the side AB AB of a triangle ABC ABC, different from A A and B B. Let P P and Q Q be the incenters of the triangles ACX ACX and BCX BCX respectively, and let M M be the midpoint of PQ PQ. Prove that: MC>MX MC>MX.
inequalitiesgeometryincentergeometry unsolved