MathDB

1

Part of 2012 IberoAmerican

Problems(2)

Ibero American 2012 - Problem 1

Source: Ibero American 2012

10/2/2012
Let ABCDABCD be a rectangle. Construct equilateral triangles BCXBCX and DCYDCY, in such a way that both of these triangles share some of their interior points with some interior points of the rectangle. Line AXAX intersects line CDCD on PP, and line AYAY intersects line BCBC on QQ. Prove that triangle APQAPQ is equilateral.
geometryparallelogramgeometry proposed
Ibero American 2012 - Problem 4

Source:

10/3/2012
Let a,b,c,da,b,c,d be integers such that the number a−b+c−da-b+c-d is odd and it divides the number a2−b2+c2−d2a^2-b^2+c^2-d^2. Show that, for every positive integer nn, a−b+c−da-b+c-d divides an−bn+cn−dna^n-b^n+c^n-d^n.
inductionmodular arithmeticstrong inductionalgebra proposedalgebra