[/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 Rounak_iitr
Let 
be an acute triangle with an altitude 
. Let 
be the orthocenter of triangle . The circle 
passes through the points 
and 
, and touches the line 
. Let 
be the diameter of . The lines~
and 
intersect for the second time at points 
and 
, respectively. The lines 
and 
intersect at the point~
. Prove that 
.
be an acute triangle with an altitude 
. Let 
be the orthocenter of triangle . The circle 
passes through the points 
and 
, and touches the line 
. Let 
be the diameter of . The lines~
and 
intersect for the second time at points 
and 
, respectively. The lines 
and 
intersect at the point~
. Prove that 
.
are cyclic.
and 
are cyclic.
a desired
and 
are similar
and 
are also similar, proving the equality.
and 
then 
