[/quote]
,
Prove that
Let 
Prove that
,
,
and 
there exists infinitely many positive integers 
and 
are not relatively primes.
or 
tile.
be an integer. Dominoes are placed on an 
board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some 
such that each row and each column has a value of 
.
be a cyclic quadrilateral inscribed in circle 
whose diagonals meet at 
.
and 
meet at 
.
intersects 
.
and 
,
and 
.
and 
concur with the tangent to 
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 an odd prime number. Suppose 
and 
are polynomials with integer coefficients such that 
,![\[
1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 +
(p-1)x}}}=\frac{P(x)}{Q(x)}.
\]](http://latex.artofproblemsolving.com/b/c/a/bca9034e2affa1d80b167451d77434fbcd8bd88a.png)
Show that all coefficients of 
other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the 
people adjacent to him and between them there are at least 
friendships of knights that are not adjacent to each other.
be 
integers which satisfy![\[ a_{n+5}+a_n>a_{n+2}+a_{n+3}\]](http://latex.artofproblemsolving.com/8/8/b/88bbb2f304525a3261851571d34863bf4d4a1e51.png)
for all integers 
.
,
lies on side 
lies on side 
.
and 
,
is parallel to 
be a point on line 
,
.
be the point on line 
,
.
,
be the circumcircle of a triangle 
.
and 
meets the sides 
and 
at 
and 
,
and 
meet 
and 
,
at 
and 
,
and 
meet at 
is the orthocenter of an acute triangle 
meets 
respectively. 
meets 
at 
at 
are collinear.
Y by mathematicsy, Adventure10, Mango247, and 5 other users
Let 
be an integer greater than 
. For a positive integer 
, let 
. Suppose that there exists a 
-element set 
such that
(a) each element of is an -element subset of 
;
(b) each pair of elements of shares at most one common element;
and
(c) each element of is contained in exactly two elements of .
Determine the maximum possible value of in terms of .
be an integer greater than 
. For a positive integer 
, let 
. Suppose that there exists a 
-element set 
such that(a) each element of
is an -element subset of 
;(b) each pair of elements of
shares at most one common element;and
(c) each element of
is contained in exactly two elements of .Determine the maximum possible value of
in terms of .
be the elements of 
.
there is exactly one, unique, pair of 
such that 
(second and third condition), there are possibly some empty intersections, so the number of intersection is not less then the number of elements of 
.
or 
.
is attainable. We construct 
as follows. 
.
then for 
we take an element on the 
-
columns and 
rows. Let each of the 
marked squares, and each column has 
marked squares; and it's easy to see that each pair of elements in 
.
-
,
-
squares, and mark the diagonal downwards again - keep doing this, and some simple computations show that when you finish you just finish filling the table.
.
.
.
elements of 
,
.
edges, so we label the edges with distinct elements of 
.
,
,
.
.
matrix with all entries either 
or 
iff 
.
nice if 
and 
.
.
and 
according to problem condition 
,
ways. Then, according to condition 
,
.
for ![\[\displaystyle A_i = \left\{ \begin{array}{lr} \{x:x=(p-1)(2n-1)+q \cup (n-1)(2n-1)+i \text{ for } 1 \leq p \leq n \text{ and } 1 \leq q \leq 2\} & \text{when}\ \ 1 \leq i \leq n-1 \\ \\ \{x:x=a(2n-1) \cup (n-1)(2n-1)+b \text{ for } 1 \leq a \leq n \text{ and } n \leq b \leq 2n-1\} & \text{when}\ \ i=n \\ \\ \{x:x=(i-n-1)(2n-1)+j \text{ for } 1 \leq j \leq 2n-1\} & \text{when}\ \ n+1 \leq i \leq 2n \end{array} \right.\]](http://latex.artofproblemsolving.com/a/b/f/abfc805380904043b2fa2708581ac481ab0279ec.png)
Then one can easily see that this set 
,
.
:
is indeed attainable and is the desired maximum. 
incidence matrix, where the rows represent the pair of 
and the column 
represents the number 
cell. Note that for each element, there is a pair of set and the intersection of any set is atmost 1. So we get 
,
.
My construction is one that I have not seen on the thread 
where 
.
and 
and using condition (b), we get that 
.
and using (c), we get that 
.
.
of triples ![\[(A_i, A_j, x \in A_i\cap A_j)\]](http://latex.artofproblemsolving.com/1/d/9/1d9b34f0ddb3f73fb4238835c52ec7f68642992e.png)
gives us 
,
.
.
or just 
.
,
built, then the idea is that 
will have the k-th element repeated from each of the 
.
