[/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 
,
such that 
and 
are tangent to 
intersects 
and the line 
.
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.
be tangent to the side 
.
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 Adventure10
Let 
be a point in the interior of triangle 
and let the lines 
meet 
at 
respectively. Let the circles with diameters 
and 
intersect at points 
; circles with diameters 
and 
intersect at points 
; and circles with diameters 
and 
intersect at points 
. Prove that 
lie on a common circle.
be a point in the interior of triangle 
and let the lines 
meet 
at 
respectively. Let the circles with diameters 
and 
intersect at points 
; circles with diameters 
and 
intersect at points 
; and circles with diameters 
and 
intersect at points 
. Prove that 
lie on a common circle.
