[/quote]
,
,
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 
.
and 
be two circles with respective centres 
and 
intersecting in two distinct points 
such that 
is an obtuse angle. Let the circumcircle of 
intersect 
and 
.
intersect 
intersect 
are concyclic.
be an acute scalene triangle with circumcenter 
.
be the feet of the altitudes from 
to 
respectively. Let 
.
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 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
,
and 
are both divisible by 
be an odd prime. Define 
over the positive integers as follows:
where 
such that 
.
such that 
and 
is a perfect square.
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?
students, each pair of students have watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
be non-negative real numbers such that 
Prove that
and 
.
Y by Adventure10, Mango247
In 
is the foot of the perpendicular from 
to the angle bisector of 
. 
and 
are respectively the feet of perpendiculars from and 
to the bisector of 
. Let 
be the intersection of lines 
and 
. Let 
be the intersection of lines 
and 
. Let 
be the intersection of lines 
and . Prove that lines 
and are parallel.
is the foot of the perpendicular from 
to the angle bisector of 
. 
and 
are respectively the feet of perpendiculars from and 
to the bisector of 
. Let 
be the intersection of lines 
and 
. Let 
be the intersection of lines 
and 
. Let 
be the intersection of lines 
and . Prove that lines 
and are parallel.
is the midline, hence 
.
is the midline. Using Cheva's tehorem in triangle 
,
hence 
.
