[/quote]
,
that verify the following equation for all 
:
is a prime number
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 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
and 
.
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 
.
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 
passing through 
of 
,
of 
.
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 
.
Y by Adventure10, Mango247
Let be given two parallel lines 
and 
, and a circle not intersecting . Consider a variable point 
on the line . The two tangents from this point to the circle intersect the line at 
and 
. Let 
be the line through the point and the midpoint of the segment 
. Prove that all the lines (as varies) have a common point.
and 
, and a circle not intersecting . Consider a variable point 
on the line . The two tangents from this point to the circle intersect the line at 
and 
. Let 
be the line through the point and the midpoint of the segment 
. Prove that all the lines (as varies) have a common point.
touch the circle in 
respectively. Also, let 
be the midpoint of 
,
,
the intersections between the circle and 
.
is a harmonic quadrilateral, so the pole of 
lies on 
.
,
.
is the pole of 
to the circle intersect the line 
;
,
,
.
s
is 
; the equations of the fixed lines 
with 
and 
respectively ; the mobile point is 
.
of the tangents to the circle 
.
,
of the points 
are the roots of the equation 
,
.![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
has the co-ordinates 
and 
.
,
.
,
be the equation of the circle 
,
and 
The equation of the polar of the point 
w.r.t. the circle 
,
to the circle 
w.r.t. the circle 
The equation of the conic 
in the points 
; the points 
,
for which 
and 
.
results that 
.
the fixed point 
be an isosceles triangle 
.
and 
.
define the points 
,
for which 
and 
.
and 
are inscribed 
,
is isosceles and 
,
and a point 
.
and 
so that 
and 
.
,
be the points for which 
and 
.
and 
.
with the rays 
