[/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 
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 
and 
are 
and 
,
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 Adventure10
Let 
be the acute triangle such that 
Let 
be the intersection of 
and angle bisector of 
Let 
be the foot of altitude from 
to 
Let 
be the intersection of circumcircle of 
and 
respectively. Prove that the lines 
is concurrent.
be the acute triangle such that 
Let 
be the intersection of 
and angle bisector of 
Let 
be the foot of altitude from 
to 
Let 
be the intersection of circumcircle of 
and 
respectively. Prove that the lines 
is concurrent.
,
and 
are perpendicular to 
and 
,
,
,
,
bisects 
,
