such that
be a positive integer. Find, in terms of 
with integer coefficients such that for every integer 
,
such that 
.
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 
is selected inside acute 
so that 
and 
.
is chosen on ray 
so that 
.
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 
.
and 
be positive integers. The cells of an 
grid are colored amber and bronze such that there are at least 
amber cells and at least 
bronze cells. Prove that it is possible to choose 
chosen cells lie in the same row or column.
and 
are erected outside an acute triangle 
Suppose that ![\[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\]](http://latex.artofproblemsolving.com/3/3/7/3378f0c4409d14bc0afddf2616992c13f624be3b.png)
Prove that lines 
and 
are concurrent.
-
of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let 
and 
?
of identical dominoes is a maximal grid-aligned configuration on the board if 
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: 
be the number of distinct maximal grid-aligned configurations obtainable from 
Y by Adventure10, Mango247
Let 
and 
be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural 
, at least one of the numbers 
or 
is an perfect square. Prove that 
, for any natural .
and 
be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural 
, at least one of the numbers 
or 
is an perfect square. Prove that 
, for any natural .
and 
be the first term, and common differences of both sequences. Using the identity, 
,
and,
Now, we concentrate on the common second terms above. Noting that 
and 
,
which is a constant.
is a perfect square, for only finitely many values of 
.
to be this sequence, and write 
with 
.
;
However, since 
as 
grows, too; which is a contradiction.
is a perfect square, for only finitely many values of 
.
since 
.
;
must hold. From here, we get 
;
