[/quote]
,
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 
,
and 
are tangent to 
intersects 
and the line 
at 
be the second point of intersection between 
and 
,
,
.
and 
the reflections of 
,
be the midpoint of 
and 
are perpendicular.
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
be the circle with diameter 
be the intersection of 
and 
be the intersection of 
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.
.
on 
on 
lies on the circumcircle of triangle 
.
and 
and 
are on the same side of 
be midpoint of segment 
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.
.
meets the side 
and 
are 
and 
and 
.
is a rectangle if and only if 
.
Y by
The exscribed circle 
of the triangle 
touches the side 
and the extensions of the sides 
and 
at the points 
, and 
, respectively, and the exscribed circle 
touches the side 
and the extensions of the sides and at the points 
, and 
, respectively. Let 
, 
. Prove that the points 
and 
lie on the same line.
of the triangle 
touches the side 
and the extensions of the sides 
and 
at the points 
, and 
, respectively, and the exscribed circle 
touches the side 
and the extensions of the sides and at the points 
, and 
, respectively. Let 
, 
. Prove that the points 
and 
lie on the same line.
By well known lengths associated with excircle and equal tangents:
Then, using the formula of a line as 
,
The point that satisfies both these equations is:
Similarly,
The point that satisfies both of these is
Now, to show that 
and 
and we are done.
