MathDB

7

Part of 2009 IMO Shortlist

Problems(3)

IMO Shortlist 2009 - Problem A7

Source:

7/5/2010
Find all functions ff from the set of real numbers into the set of real numbers which satisfy for all xx, yy the identity f(xf(x+y))=f(yf(x))+x2 f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2
Proposed by Japan
functionalgebrafunctional equationIMO Shortlist
IMO Shortlist 2009 - Problem G7

Source:

7/5/2010
Let ABCABC be a triangle with incenter II and let XX, YY and ZZ be the incenters of the triangles BICBIC, CIACIA and AIBAIB, respectively. Let the triangle XYZXYZ be equilateral. Prove that ABCABC is equilateral too.
Proposed by Mirsaleh Bahavarnia, Iran
geometryincenterreflectioninequalitiesIMO Shortlist
IMO Shortlist 2009 - Problem N7

Source:

7/5/2010
Let aa and bb be distinct integers greater than 11. Prove that there exists a positive integer nn such that (an−1)(bn−1)(a^n-1)(b^n-1) is not a perfect square.
Proposed by Mongolia
number theoryPerfect SquareIMO Shortlistexponential