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.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
+1 w
be real numbers such that 
Prove that 
Where 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.
+1 w
, prove that there exists a positive integer 
, such that for any integer 
, 
can be expressed by a sum of positive integers 
's as![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.
equally spaced points on a circular track 
of circumference . The points are labeled 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
. She hops along from to 
, then from to 
, until she reaches 
, after which she hops back to . When hopping from 
to 
, she always hops along the shorter of the two arcs 
of ; if 
is a diameter of , she moves along either semicircle. arcs which Bunbun traveled, over all possible labellings of the points.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and .
board with a number written in each cell. Every two neighbour rows sum up to at least 
, and every two neighbour columns sum up to at most 
. Find the sum of all numbers on the board.
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
with no parallel sides is inscribed in a circle 
. Circles 
are inscribed in triangles 
, respectively. Common external tangents are drawn between 
and 
, and 
, and 
, and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle 
. Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.
and 
, 
, intersect at 
and 
such that 
. The line 
meets 
at 
and 
, and 
at 
and 
(in the order , , , ). The line 
meets at 
and 
at 
, and the line 
meets at 
and 
at 
. Prove that![\[ \frac{ r_2}{r_1} = \frac{KE}{KM} \cdot \frac{LN}{LD} . \]](http://latex.artofproblemsolving.com/7/4/c/74cb039811c34f066ab9814c8a17aa4c93c6002c.png)
and , , intersect at and such that . The line meets at and , and at and (in the order , , , ). The line meets at and at , and the line meets at and at . Prove that
and 
, 
, intersect at 
and 
such that 
. The line 
meets 
at 
and 
, and 
at 
and 
(in the order , , , ). The line 
meets at 
and 
at 
. The line 
meets at 
and 
at 
. Prove that 
. 
The division 
is harmonically\ ;
The angles 
, 
, 
, 
, 
hava same measure, 
.
(see Camaronsou's) 
.[/size]
are isosceles, if 
is the intersection of 
and 
and 
, and also 
so
and 
and we want to prove that
which is equivalent with
, note that
Prove that 
