[/quote]
,
for which the inequality ![\[ (a^2+3)(b^2+3)(c^2+3)(d^2+3) + k(a-1)(b-1)(c-1)(d-1) \ge 0 \]](http://latex.artofproblemsolving.com/4/d/b/4db42ebabc7309bba9629a9a7125be43abf113f0.png)
holds for all real numbers 
,
,
,
.
such that 
.
at points 
respectively. Let 
and 
be points on the segment 
respectively such that 
is parallel to 
.
be the circle that is externally tangent with the excircle
.
,
,
be an odd prime and 
is a bijection and 
is an integer such that 
Show that 
is a multiple of 
,
be the incenter and the 
-
;
,
be the touching points of the incircle with 
,
respectively; 
be the common point of 
and 
.
from 
at point 
.
is given. Over the side 
on the outside. The midpoint of the segment 
is 
and the midpoint of the side 
is 
.
.
and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching 
without passing through 
)
and orthocenter 
be the midpoint of 
intersects 
,
intersects 
.
and 
is a parallelogram.
is tangent to the circumcircle of triangle 
.
have the property that both![\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]](http://latex.artofproblemsolving.com/d/e/c/dec527c4673470e7023d6aed00fac90268bd2fcc.png)
are consecutive perfect squares.
and 
are given. Let 
be arbitrary points on 
.
.
,
lies on the circumcircle of 
.
be a point inside a quadrilateral 
.
,
,
are isogonally conjugated to 
,
,
,
concur.
respectively touch the extension of 
meets 
,
and 
is right
Y by
Let 
be a positive integer and 
be a family of at most 
distinct subsets of the set 
with the following property: we can consider distinct points in the plane, labelled 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to subsets in for any 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
be a positive integer and 
be a family of at most 
distinct subsets of the set 
with the following property: we can consider distinct points in the plane, labelled 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to subsets in for any 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
if and only if there is segment between 
.
be a number of subsets, which contain 
for some integers 
and 
.
from our first sentence. This means that all 
,
have to pairwise different. This means that 
,
,
for subsets 
(because 
)
.
