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.
chess board. The allowed move is to add an integer 
to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.
be positive integers such that ![$\sqrt[n]{a}+\sqrt[n]{b}$](http://latex.artofproblemsolving.com/d/a/a/daaf95cd3381060559529aa4a318e97b727eac31.png)
is an integer. Prove that 
are both the 
th power of 
positive integers.
and 
Show that
Where 
and 
Show that
Where
be positive integers such that 
is not squarefree. Prove that there exist a positive real 
, such that 
and 
are coprime for any positive integer .
be a quadrilateral inscribed in a circle 
. The lines 
and 
meet at 
, the lines 
and 
meet at 
, and the diagonals 
and 
meet at 
. Let be the midpoint of the segment 
, and let 
be the common point of the segment 
and the circle . Prove that the circumcircle of the triangle 
and are tangent to one another.
other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the people adjacent to him and between them there are at least 
friendships of knights that are not adjacent to each other.
, call all positive integers that cannot be written as 
(where 
are non-negative integers) bad, and define 
to be the sum of all bad numbers raised to the power of 
. Prove that there exists a positive integer , such that for any 
as described, 
divides 
.
In the 
plane with the origin 
, the line 
intersects the lines 
, and 
at the points 
, respectively. Find the maximum value of the area of 
plane we have the points 
. The area of 
as a function of 
is represented by:
(i);
(ii) (since 
);
(a maximum).
.
. Prove that:
Show that
Where 
Where 
and 
Prove that
