[/quote]
,
be an acute scalene triangle with circumcenter 
.
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 orthocenter of 
and 
be the lines through 
respectively. Let 
be the intersection of 
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 
,
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
In the trapezoid 
, the base 
is smaller than the 
base. The point 
is chosen such that 
is parallel to BC and 
is parallel to 
. The points 
and 
are chosen on the and rays respectively, such that 
and 
.
(a) Show that the lines
and 
go through the same point.
(b) Assuming that the circumscribed circumferences of the
and 
triangles intersect at two points, show that one of those points belongs to the line .
, the base 
is smaller than the 
base. The point 
is chosen such that 
is parallel to BC and 
is parallel to 
. The points 
and 
are chosen on the and rays respectively, such that 
and 
.(a) Show that the lines
and 
go through the same point.(b) Assuming that the circumscribed circumferences of the
and 
triangles intersect at two points, show that one of those points belongs to the line .This post has been edited 2 times. Last edited by parmenides51, Sep 2, 2022, 5:50 AM
