[/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 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
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 
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 the midpoint of side BC of an acute-angled triangle 
. Let 
and 
be the center of the excircle of triangle 
tangent to side 
and the center of the excircle of triangle 
tangent to side 
, respectively. The circumscribed circle of triangle 
intersects line
for the second time at point 
, and the circumcircle of triangle 
is at point 
. Prove that 
.
be the midpoint of side BC of an acute-angled triangle 
. Let 
and 
be the center of the excircle of triangle 
tangent to side 
and the center of the excircle of triangle 
tangent to side 
, respectively. The circumscribed circle of triangle 
intersects line
for the second time at point 
, and the circumcircle of triangle 
is at point 
. Prove that 
.
.
.
Thus, 
Then by LoS on 
,
and similarly 
,
After simplifying using product-sum, we want 
But from LoS on 
,
we have 
,
