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
Let 
be a prime number. We construct a directed graph of vertices, labeled with integers from 
to 
. There is an edge from vertex 
to vertex 
if and only if 
. Let 
denotes the length of the longest directed cycle in this graph. Prove that can attain arbitrarily large values.
be a prime number. We construct a directed graph of vertices, labeled with integers from 
to 
. There is an edge from vertex 
to vertex 
if and only if 
. Let 
denotes the length of the longest directed cycle in this graph. Prove that can attain arbitrarily large values.This post has been edited 1 time. Last edited by mathuz, Dec 28, 2023, 8:09 AM
and let 
for all 
.
,
for all positive integers 
.
,
,
,
,
is minimal. Then 
and we will show that 
for all positive integers 
doesn't have a 
term, so we have 
,
it's 
,
by induction again it's the same, so we are done.
