AoPS Academy
be reals such that 
and 
Prove that
Let be reals such that and 
Prove that
Let be reals such that and 
Prove that
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
, for 
in at least 
different ways.
and colors are given. We will say that a set of 
points on a plane is 
, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of the least integer such that: in every set of 
points of a plane, no three of which are collinear, consisting of 
points of every color there exists a subset.
?
inside the convex quadrilateral 
such that the triangles 
, 
, 
, 
have the same area, then one of the diagonals of bisects the area of the quadrilateral.
doesn't lie on 
(if it did, we'd be done). Let 
and 
. Clearly, 
are both midpoints of , so lies on 
, and we are done.
