[/quote]
,
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?
is defined by 
and for all positive integers 
,
.
,
such that 
is a multiple of 
.
be a subset of 
.
.
has circumcircle 
and incenter 
.
at 
respectively. Circumcircle of triangle 
intersects 
distinct from 
,
intersects 
distinct from 
.
points 
are concyclic.
.
is a parallelogram. Prove that 
.
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
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.
such that 
.
that verify the following equation for all 
:
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
that satisfy the equation 
.
Y by
One night, three flying saucers landed on the field, red, green and blue. Each of them densely covered a certain triangular area with a powder of its color. The next morning, surprise villagers found that the intersection of red and green has the shape of a triangle, the intersection of red and blue is quadrangular, and the intersection of green and blue is pentagonal. Can the intersection of all three sections be hexagonal?
This post has been edited 2 times. Last edited by parmenides51, Mar 4, 2021, 12:52 AM
