MathDB

3

Part of 2007 International Zhautykov Olympiad

Problems(2)

infinity of positive integers satisfying...

Source: Third Zhautykov Olympiad, Kazakhstan, 2007

4/25/2007
Show that there are an infinity of positive integers nn such that 2n+3n2^{n}+3^{n} is divisible by n2n^{2}.
inductionmodular arithmeticnumber theory proposednumber theory
super nice 6-th problem, an interesting result.

Source: Third Zhautykov Olympiad, Kazakhstan, 2007

4/25/2007
Let ABCDEFABCDEF be a convex hexagon and it`s diagonals have one common point MM. It is known that the circumcenters of triangles MAB,MBC,MCD,MDE,MEF,MFAMAB,MBC,MCD,MDE,MEF,MFA lie on a circle. Show that the quadrilaterals ABDE,BCEF,CDFAABDE,BCEF,CDFA have equal areas.
geometrycircumcircletrapezoidtrigonometrygeometric transformationhomothetyratio