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.
other than a polynomial satisfying
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.
.
. which are not perfect squares. Calculate the 
-th term of the sequence.
, let 
be the midpoint of side 
and 
be the midpoint of segment 
. Lines 
and 
intersect at 
. Show that 
. such that 
is a square perfect.
friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?
games and lost games; there were no ties. How many sets of three teams 
were there in which 
beat 
, beat 
, beat and beat 
games and lost games; there were no ties. How many sets of three teams were there in which beat , beat , beat and beat 
Prove that
Where 
and 
Prove that
Prove that
![\[\begin{cases}
3x^2-2xy+3y^2+\dfrac{2}{x^2-2xy+y^2}=8\\
2x+\dfrac{1}{x-y}=4
\end{cases}\]](http://latex.artofproblemsolving.com/e/b/f/ebf41764f74c98da66dc31c48975f27bd734c334.png)
?