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.
and 
then find the value of 
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.
be the number of positive integers such that divides 
and 
divides
where 
denotes the number of positive integers that divide , including 
and . Find the remainder when is divided by 
.
and 
Show that
Where 
and 
Show that
Where
consists of the union of disjoint pieces. Compute the distance between the endpoints of any one piece
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.
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
![\[ i tried x = k + r where k and r are floor and fractional parts 0 \leq r < 1 \]](http://latex.artofproblemsolving.com/3/2/4/324d37d221a373a03d15c358ddf9b72cbbdd11fd.png)
then solved the quadratic equation in terms of k
isnt equal to 
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which 
? or 
.
, we obtain the inequalities 
and 
.
or 
,
is equal to 
or 
. Let us consider 5 cases:
so 
, and only 
works.
so 
, but none of them works.
so 
, and only 
works.
so 
, and only 
works.
so 
, and only 
works.
.
.
.
Let 
.
Show that
Where 
Where 
![\[\lfloor x \rfloor = x\]](http://latex.artofproblemsolving.com/4/a/0/4a0e191f2cd148199766b7aa190de609551563a1.png)
, The expression is 
,
![\[\lfloor x \rfloor = x - 1\]](http://latex.artofproblemsolving.com/3/4/c/34c53224b71736587cd83dc75c94a9bfe9abd6bb.png)
.
.
and 
.
.