[/quote]
,
is a prime number
.
be the feet of the altitudes from 
to 
respectively. Let 
be the circle with diameter 
.
be the intersection of 
and 
.
and 
be the lines through 
tangent to 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
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 
beyond 
,
and 
.
is a rectangle if and only if 
.
satisfying
Prove
is a positive integer. Call all positive divisors of 
and 
beautiful divisors and difference of any 
that verify the following equation for all 
:
passing through 
and 
passing through 
are tangent at 
,
.
and 
meet at 
.
be a point such that 
is tangent to 
is tangent to 
are collinear.
.
When does equality hold?
denote the set of all positive integers less than 
that are relatively prime to 
.
reducible over the rational numbers, given that it has integer coefficients?
blocks, each weighing at least 
.
with 
you can choose a subset of the blocks whose total weight is at least 
.
such that 
.
Y by
Let 
be an acute triangle with 
. Let 
be the circle circumscribed to the triangle and 
the midpoint of the smaller arc 
of this circle. Let 
be the incenter of and let 
and 
be points on sides 
and 
, respectively, such that 
and lies on the segment 
. Let 
be the second intersection point of the circumcircle of the triangle 
with with 
. Let 
and 
be the intersection points of the lines 
and 
with different from , respectively. Let 
and 
be the intersection points of lines 
and 
with lines AB and , respectively. Show that the line 
passes through the midpoint of .
be an acute triangle with 
. Let 
be the circle circumscribed to the triangle and 
the midpoint of the smaller arc 
of this circle. Let 
be the incenter of and let 
and 
be points on sides 
and 
, respectively, such that 
and lies on the segment 
. Let 
be the second intersection point of the circumcircle of the triangle 
with with 
. Let 
and 
be the intersection points of the lines 
and 
with different from , respectively. Let 
and 
be the intersection points of lines 
and 
with lines AB and , respectively. Show that the line 
passes through the midpoint of .
