[/quote]
,
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 
respectively. Let 
be the intersection of 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
and orthocenter 
is constant.
be tangent to the side 
.
on 
and 
on 
and 
,
lies on the circumcircle of triangle 
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
.
and 
on 
and 
are on the same side of 
be midpoint of segment 
with 
and 
that verify the following equation for all 
:
be positive real integer such that 
Prove that
such that 
and 
is a prime number
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 
and 
.
is a rectangle if and only if 
.
satisfying
Prove
and 
beautiful divisors and difference of any 
that verify the following equation for all 
:
Y by
Point 
is the midpoint of side 
of a non-isosceles triangle 
, and point 
is the center of a circle around it. Circle with 
diameter intersects for second time the sides 
and 
at points 
and 
, respectively. Point 
is midpoint of line 
. Prove that lines 
and 
are parallel.
is the midpoint of side 
of a non-isosceles triangle 
, and point 
is the center of a circle around it. Circle with 
diameter intersects for second time the sides 
and 
at points 
and 
, respectively. Point 
is midpoint of line 
. Prove that lines 
and 
are parallel.
.
,
,
.
