[/quote]
,
such that for any real numbers 
and 
,![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq f\left((x+y)^2\right)
\]](http://latex.artofproblemsolving.com/2/0/0/20053f093b3621cbab571e5b635a3ae64573280f.png)
Proposed by Anton Trygub
is filled with the numbers 
,
times. Show that there is a row or column with at least 
distinct numbers.
for which there exists a positive integer 
with the property that 
is divisible by 
.
Prove that
Where 
scribable if there exists a triangle with circumcircle 
and incircle 
.
scribable pairs.
cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
and 
Prove that
be positive reals such that 
.![\[(a-b)^2 + a + b \geq 2\]](http://latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
Prove that
be a polynomial with real coefficients such that all its roots are real and nonzero. Prove that there are no 2 consecutive coefficients both zeros.
(where 
are the digits and 
)
For example the permutation sum of 20253 is 
Let 
and 
.
Y by
In a triangle 
, 
is the orthogonal projection of the incenter 
onto 
. Line 
meets the incircle again at 
. Line 
intersects side at point 
. Suppose that the segment IO is parallel to , where 
is the circumcenter of 
. If 
is the circumradius and 
the inradius of the triangle, prove that 
.
, 
is the orthogonal projection of the incenter 
onto 
. Line 
meets the incircle again at 
. Line 
intersects side at point 
. Suppose that the segment IO is parallel to , where 
is the circumcenter of 
. If 
is the circumradius and 
the inradius of the triangle, prove that 
.
