[/quote]
,
,such that 
P
with integer coefficient that satisfy the following
for each 
,
that satisfy the equation
be reals such that 
Prove that
Let 
Prove that
with two altitudes 
and 
.
be the point on the line 
.
and 
intersect at point 
,
is the point on segment 
such that 
.
is isosceles.
of positive integers with ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
.
be a connected graph and let 
be two disjoint subsets of its vertices, such that there are no edges between them. Given that 
has 
has 
?
,
,
.
be the circle which touches 
,
lies outside of the excentral triangle of 
.
and 
cyclically. Let 
be the circle externally tangent to 
to be the least positive integer for which the following holds:
into 
disjoint subsets 
,
.
for which![\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]](http://latex.artofproblemsolving.com/2/0/9/2092eb944c746d1449e7865f9da4e0065b75bdef.png)
so that 
for all 
.
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 
.
as 
,
built, then the idea is that 
will have the k-th element repeated from each of the 
.
