[/quote]
,
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
that verify the following equation for all 
:
is a prime number
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
such that 
.
be a quadrilateral inscribed in a circle 
be a point on the extension of 
such that 
and 
are tangent to 
intersects 
and the line 
at 
.
be the second point of intersection between 
and 
,
be a triangle with incenter 
,
and 
at 
.
and 
the reflections of 
with 
,
and 
are perpendicular.
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
be the intersection of 
and 
be the lines through 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
is constant.
be tangent to the side 
.
on 
on 
lies on the circumcircle of triangle 
.
and 
and 
are on the same side of 
with 
and 
be positive real integer such that 
Prove that
such that 
and 
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
.![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
Y by
Inside the circle 
there are circles 
and 
intersecting at points 
and 
and tangent to the circle at points 
and 
. It turned out that points 
and lie on one straight line. Find the radius of the circle if the radii of the circles and are 
and 
, respectively.
there are circles 
and 
intersecting at points 
and 
and tangent to the circle at points 
and 
. It turned out that points 
and lie on one straight line. Find the radius of the circle if the radii of the circles and are 
and 
, respectively.This post has been edited 1 time. Last edited by parmenides51, Jan 23, 2021, 8:23 PM
.
