[/quote]
,
be an acute scalene triangle with circumcenter 
.
be the feet of the altitudes from 
to 
respectively. Let 
.
be the circle with diameter 
be the intersection of 
and 
be the orthocenter 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 
and 
are 
and 
beyond 
and 
and 
.
is a rectangle if and only if 
.
is a prime number
is a positive integer. Call all positive divisors of 
and 
beautiful divisors and difference of any 
such that 
.
is defined by 
and for all positive integers 
,
.
,
such that 
is a multiple of 
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
be an odd prime. Define 
over the positive integers as follows:
where 
such that 
.
such that 
and 
is a perfect square.
be the nonnegative integers in 
where 
.
.
.
where 
,
?
where 
,
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
such that 
divides 
.
Y by
On the circle 
the points 
and 
are chosen so that the tangents to the circle on the points 
and 
and the line 
intersect at point 
. On the lines 
and 
, the points 
and 
are chosen, respectively, so that the line 
passes through point and 
. Prove that 
.
the points 
and 
are chosen so that the tangents to the circle on the points 
and 
and the line 
intersect at point 
. On the lines 
and 
, the points 
and 
are chosen, respectively, so that the line 
passes through point and 
. Prove that 
.
