interesting if 
is composite and for every divisor 
of 
and 
is also a divisor of 
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 a triangle ABC. A1, B1 and C1 are the points of contact of the inner circumcircle of the triangle with the sides BC, AC and AB respectively. The point of contact of AA1 with B1C1 and the circumcircle are called L and Q respectively. M is the midpoint of B1C1. The point of intersection of lines BC and B1C1 is called T. P is the foot of the perpendicular drawn to AT from point L. Show that points A1, M, Q and P lie on a circle.
