AoPS Academy
, the points 
and 
are taken such that the point lies between the points 
and , and 
. On segments 
and 
, points 
and 
are taken so that 
and 
. Prove that 
.
and points and on sides 
and 
, respectively, such that 
. Let 
and 
be intersection points of lines 
and 
with circumcircle 
of triangle 
, and 
and 
intersection points of lines 
and 
with side 
. If 
, prove that circle contains intersection point of angle bisector of 
with
such that 
divides 
for all positive integers 
and 
with 
.
such that 
Follows for all reals 
.
denote the set of positive integers. Find all functions 
such that for positive integers and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
be an integer. Rowan and Colin play a game on an 
grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is orderly if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of 
, how many orderly colorings are there? is selected inside acute 
so that 
and 
. Point is chosen on ray 
so that 
. Let be the midpoint of . is tangent to the circumcircle of triangle 
.
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
are real numbers and is defined as![\[
f\left( x\right) =\left| x-a_{1}\right| +\left| x-a_{2}\right|
+\ldots+\left| x-a_{n}\right| ,
\]](http://latex.artofproblemsolving.com/1/2/a/12ad7d81cfbe2a5051a30a72c642618dff94df04.png)
is first decreasing, and then increasing. More precisely, if is odd, say 
then is strictly decreasing on the interval ![$(-\infty,a_{k}]$](http://latex.artofproblemsolving.com/8/0/0/80059c26862f71b2676373167a9bdde0ba84a6e3.png)
and strictly increasing on 
, while if is even, say 
then is strictly decreasing on the interval ![$(-\infty,a_{k}],$](http://latex.artofproblemsolving.com/a/5/b/a5b8693454345f2a08ecdfa658098cc7f539f5b7.png)
constant on ![$[a_{k},a_{k+1}]$](http://latex.artofproblemsolving.com/8/b/b/8bb9077f24dc8489ab5a63e97d770fa05928acb5.png)
and strictly increasing on 
be our arithmetic sequence and defined as above. We see that 
, hence three values of are equal. This is possible only if is even (say 
) and ![$0,-1,2\in\lbrack a_{k},a_{k+1}].$](http://latex.artofproblemsolving.com/4/5/c/45c822a73669b9a2d73870a30c660569eaceac41.png)
We deduce that the common difference 
is at least 
and 
A short computation shows that in this case![\[
507=\sum\left| a_{i}\right| \geq3k^{2},
\]](http://latex.artofproblemsolving.com/a/6/5/a65627fe06406ede61c2444d94052851362c8a94.png)
that is, 
,
and 
be an arithmetic progression, 
,![\[ \sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507. \]](http://latex.artofproblemsolving.com/7/7/2/7723ed97373f545f14d0b46c0034dab7d7daecf4.png)
,
,
or more than 
)
,