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.
be the circumcenter of an acute-angled triangle 
.
meets the side 
at 
.
and 
are 
and 
,
and 
beyond 
,
and 
such that 
and 
.
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 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
,
,
is not greater than 
.
.
.
Y by
Let 
be an equilateral triangle of area 
cm
. Points 
divide the segments ![$[AB], [BC] ,[CA]$](//latex.artofproblemsolving.com/6/9/2/692cabdc3172676027cc474fbd36490f10f0f36a.png)
, respectively, in the ratio 
. Line 
intersects the lines 
and 
respectively at the points 
and 
, and the line intersects the line at point 
. Find the area of the triangle 
.
be an equilateral triangle of area 
cm
. Points 
divide the segments ![$[AB], [BC] ,[CA]$](http://latex.artofproblemsolving.com/6/9/2/692cabdc3172676027cc474fbd36490f10f0f36a.png)
, respectively, in the ratio 
. Line 
intersects the lines 
and 
respectively at the points 
and 
, and the line intersects the line at point 
. Find the area of the triangle 
.
etc. Denote 
.
Let 
and line 
.
Van Aubel's theorem yields 
Thus 
Let 
and line 
.
Van Aubel's theorem yields 
Thus 
So
Analogously 
It's now obvious that 
