[/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
Given an acute-angled 
C with altitude 
( 
C). The perpendicular bisector of 
intersects 
at point 
. Let 
be the midpoint of 
, where 
is the foot of the perpendicular from 
on 
. Point 
is the symmetric to 
wrt . Point 
lies on the line , such that 
. Prove that 
.
C with altitude 
( 
C). The perpendicular bisector of 
intersects 
at point 
. Let 
be the midpoint of 
, where 
is the foot of the perpendicular from 
on 
. Point 
is the symmetric to 
wrt . Point 
lies on the line , such that 
. Prove that 
.
to be 
and computing everything else in Cartesian style kills it nicely!
