AoPS Academy
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 
.
. when![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
be the circumcenter of an acute-angled triangle 
with 
. The line 
meets the side 
at 
. The circumcenters of the triangles 
and 
are 
and 
, respectively. Extend the sides 
and 
beyond 
, and choose on the respective extensions points 
and 
such that 
and 
. Prove that the quadrilateral 
is a rectangle if and only if 
.
is a prime number
satisfying
Prove
that satisfy the equation 
.
be a circle, and let a line 
intersect at two points, 
and 
Circles 
and 
are internally tangent to at points 
and 
respectively, and both are tangent to at a common point 
Similarly, circles 
and 
are externally tangent to at and respectively, and are tangent to at points and 
respectively. is 
the segment 
has a length of 
and 
find the length of segment 
,
,
, 
, 
is not greater than 
.
and full surface 
.
.
. We need to show that there is a point inside whose distances from every face is at least 
.
of , consider the rectangular solid with base and height , directed inside . Call it 
.
such that there is a face 
that 
, it's not hard to show that there exists a face 
that 
. (For a rigorous argument, begin by choosing the face that minimizes the distance.) Consider the union of all 
over all faces. This forms a polyhedron of volume strictly less than 
. ( of two adjacent faces is a polyhedron.) Hence, there exists a point 
that does not lie in the union. 
is the desired point.
and 
Prove that
.
Prove that
and 
Prove that
.
.
