[/quote]
,
,
is differentiable such that 
Then which of the following is correct?
always exists but not necessarily zero.
is twice differentiable.
such that for all 
,![\[xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)\]](http://latex.artofproblemsolving.com/5/b/1/5b101ead6195155ccbd99b757016e8f73dac4411.png)
Proposed by Aritra Mondal
be positive real numbers such that 
.
prove inequality 
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
,
,
checkered board is cut into several squares with integer side length. The size of a square is it's side length.
of integers, a pair 
with 
is called interesting if there exists a pair 
of integers with 
such that 
For each 
,
.
such that, for all 
,
cards on which the numbers 
,
,
,
be the sum of the numbers written on the cards that Bob keeps. Find the greatest positive integer 
for which Bob can guarantee that 
be positive real number such that 
Find the minimum value of 
Prove that
Where 
Y by
Let 
be a triangle and 
be the foot of the bisector of 
. Let 
and 
be the circumcenters of 
and 
respectively and let 
and 
be the projections of 
and 
through the bisectors of the angles 
and 
respectively. The incircle of touches 
and 
at points 
and 
respectively and the bisectors of angles and meet the perpendicular bisector of 
at points 
and 
respectively. Prove that the five lines 
and 
are all concurrent.
be a triangle and 
be the foot of the bisector of 
. Let 
and 
be the circumcenters of 
and 
respectively and let 
and 
be the projections of 
and 
through the bisectors of the angles 
and 
respectively. The incircle of touches 
and 
at points 
and 
respectively and the bisectors of angles and meet the perpendicular bisector of 
at points 
and 
respectively. Prove that the five lines 
and 
are all concurrent.
