.
,
.
lies on 
such that 
.
passing trough 
at 
.
.
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 Mango247
In a language
an alphabet with 
letters is used
words are exactly all sequences of 
not necessarily different 
letters of length 
Two ends of a paper strip are glued so that the strip forms a ring the strip bears a sequence of 
letters
Say that a word is singular if one can cut a piece bearing exactly that word from the strip but one cannot cut out two such non-overlapping pieces It is known that one can cut out 
non-overlapping pieces each containing the same word Determine the largest possible number of singular words
(Bogdanov I.)
an alphabet with 
letters is used
words are exactly all sequences of 
not necessarily different 
letters of length 
Two ends of a paper strip are glued so that the strip forms a ring the strip bears a sequence of 
letters
Say that a word is singular if one can cut a piece bearing exactly that word from the strip but one cannot cut out two such non-overlapping pieces It is known that one can cut out 
non-overlapping pieces each containing the same word Determine the largest possible number of singular words(Bogdanov I.)
This post has been edited 1 time. Last edited by Hopeooooo, May 23, 2022, 2:19 PM
Reason: 1
Reason: 1
By a piece we always mean a piece of the strip containing exactly 
consecutive letters; Different piece may contain the same word. Say that a piece is singular if the word it contain is such. We start with constructing an example containing 
singular words. Define a word 
This will be the word having 
non-overlapping copies on the strip. There exist exactly 
possible 
l
Put them onto the strip in an arbitrary order, separating each two sequences by an instance of 
Each segment of the strip containing one 
letters. Clearly, the obtained strip contains 
non-overlapping copies of 
different positions (starting from the 
s
n
t
singular words. Consider an arbitrary piece 
containing a a word 
Either this piece contains a unique non empty prefix which coincides with some suffix of 
or there is no such prefix only in this case we will say that such prefix is empty. Let 
Notice that the defined prefix and suffix do not overlap whenever 
(if 
we have 
)
If the piece contains a part then 
and 
Thus, piece 
and in this case the position of the part at 
so this piece is singular. We have proven that the constructed example works. It remains to prove that the number of singular words cannot exceed 
Enumerate the positions in the strip successively by 
(the numeration is cyclic module 
denote the piece starting at positions 
and let 
be the word on that piece. Let 
be a positive such that the pieces 
are pairwise disjoint and contain the same word 
(from the problem statement). Clearly, those pieces are not singular. For 
and 
we say that a piece 
is a rank 
follower, while 
is a rank 
followers of all ranks, at most 
non-singular pieces, which implies the desired bound. Thus, we are left to prove the claim quoted above. For any rank 
define its tail as its suffix of length 
We regard a tail as a sequence of letters). We show by induction on 
that for every sequence 
consisting of 
letters, there are no more than 
followers whose tails contain 
The base case 
is obvious; If a follower with tail 
then every singular follower's tail starting with 
For every 
there are at most 
such followers, by the inductive hypothesis. So the total number of such followers does not exceed 
as desired. Finally, if there is a singular follower 
whose tail is 
of 
itself) is at most 
as desired again.
