Some unsolved problems January 3, 2010Posted by Damien Jiang in Problem-solving.
Tags: functional equation, geometry, IMO longlist, incircle, integer functional equation, Russian Olympiad
Happy New Year!
Since we have been too lazy to post lately (and the so-not-lazy Akhil posts mostly elsewhere now), I’m going to post some problems that I probably should be able to solve, but haven’t.
First, an interesting (and apparently classic) problem given to me by Tim Chu, USA IMO alternate:
Find all such that f is multiplicative (for relatively prime positive integers), monotonically increasing, and f(2) = 2.
A second from an old IMO longlist:
Show that the union of all segments with endpoints in includes every point in .
EDIT: Well apparently this is wrong. However, part b of the problem asks if the convex hull of is equal to .
And a third from Russia 2005:
A quadrilateral ABCD without parallel sides is circumscribed around a circle with center O. Show that O is the barycenter of ABCD iff (OA)(OC) = (OB)(OD).
(Looks quite vector/complex-number friendly, but… I probably didn’t try hard enough.)