MathDB

3

Part of 2010 Pan African

Problems(2)

f(x+f(y))=f(x)-y

Source: Pan African Olympiad 2010

10/1/2011
Does there exist a function f:Z→Zf:\mathbb{Z}\to\mathbb{Z} such that f(x+f(y))=f(x)−yf(x+f(y))=f(x)-y for all integers xx and yy?
functionalgebradomaininductionalgebra proposed
BM=CF and <MBC=<FCA means ABC is equilateral

Source: Pan African Olympiad 2010

10/1/2011
In an acute-angled triangle ABCABC, CFCF is an altitude, with FF on ABAB, and BMBM is a median, with MM on CACA. Given that BM=CFBM=CF and ∠MBC=∠FCA\angle MBC=\angle FCA, prove that triangle ABCABC is equilateral.
geometrycircumcirclegeometry proposed