people who do not know each other. Prove that it is possible to introduce them in such a way that no three of them have the same number of acquaintances.
touch 
,
and 
at points 
,
and 
respectively. The corresponding excircle touches the side 
.
be the common point of 
and the incircle, closest to 
be the common point of 
and 
.
.
be the 
-
and 
for 
)
,
be the smallest integer 
.
,
to 
: 
interesting if 
is composite and for every divisor 
of 
and 
is also a divisor of 
distinct points on a circle. Consider the set of distances between any two out of the 
such that for any positive integers 
,
we have 
.
,
is divisible by 
.
be a set of 2025 positive real numbers. For a subset 
,
as the median of 
when all elements of 
.![\[
P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.
\]](http://latex.artofproblemsolving.com/1/d/f/1df2ea8487289111939a53dcc49576fbe6fdef37.png)
Find the smallest real number 
such that for any set ![\[
P(A) - Q(A) \leq C \cdot \max(A),
\]](http://latex.artofproblemsolving.com/c/d/e/cdea343d11c722a423e857aeb01202863405773e.png)
where 
denotes the largest element in 
Y by
Given triangle 
inscribed in circle 
and circumscribed about circle 
. A circle passing through 
and 
is tangent to at 
and intersects 
and 
at 
and 
respectively. Similarly, define 
and 
correspondingly. Let 
be the triangle formed by the radical axis of circles 
, 
, 
(as shown in the figure); 
is the triangle formed by the intersection of lines 
, 
, 
(as shown in the figure).
a) Prove that: Triangle and triangle are in perspective axially.
b) Let
be the point of concurrence of 
, 
, 
. Prove that: , 
, 
are collinear.
inscribed in circle 
and circumscribed about circle 
. A circle passing through 
and 
is tangent to at 
and intersects 
and 
at 
and 
respectively. Similarly, define 
and 
correspondingly. Let 
be the triangle formed by the radical axis of circles 
, 
, 
(as shown in the figure); 
is the triangle formed by the intersection of lines 
, 
, 
(as shown in the figure).a) Prove that: Triangle
and triangle are in perspective axially.b) Let
be the point of concurrence of 
, 
, 
. Prove that: , 
, 
are collinear.Attachments:
is NOT aligned with 
and 
.
.
