AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
such that for any real numbers 
and 
, the following inequality holds:![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq f\left((x+y)^2\right)
\]](http://latex.artofproblemsolving.com/2/0/0/20053f093b3621cbab571e5b635a3ae64573280f.png)
Proposed by Anton Trygub
is filled with the numbers 
, each appearing exactly 
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 and 
is divisible by 
.
scribable if there exists a triangle with circumcircle 
and incircle 
. Prove that among distinct circles there are at most 
scribable pairs. cells is a connected figure consisting of equal-sized cells[1].
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, one with 
cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
be positive reals such that 
. Prove that ![\[(a-b)^2 + a + b \geq 2\]](http://latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
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 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and be two fivedigit integers with the same permutation sum.
.
and a positive integer 
is given.
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
. Also 
. We have 
. We will proceed by using induction. Suppose that 
holds, we will prove 
. On the other hand, for all 
Thus, we obtain the following
implying that the sequence 
must be eventually constant as desired.
. Also . We have . We will proceed by using induction. Suppose that holds, we will prove . On the other hand, for all Thus, we obtain the followingimplying that the sequence must be eventually constant as desired.
Prove that
Where 
and 
Prove that
Prove that
