[/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 
.
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 
.
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 
and 
passing through 
are tangent at 
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
We are given a mechanism that can perform the following operations:
Joining any two points of a plane by a straight line;
Constructing the reflection 
of a given point 
in a given line 
.
Given a triangle A
, using the given mechanism, construct
(a) its centroid,
(b) its circumcenter.
Joining any two points of a plane by a straight line; Constructing the reflection 
of a given point 
in a given line 
.Given a triangle A
, using the given mechanism, construct(a) its centroid,
(b) its circumcenter.
