[/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 
and 
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 
respectively. Let 
be the intersection of 
and 
.
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 Eagleka94, Adventure10, Mango247
In a cyclic quadrilateral 
whose largest interior angle is 
, lines 
and 
intersect at point 
, while lines 
and 
intersect at point 
. A point 
is taken in the interior of quadrilateral for which 
. 
is the circumcenter of quadrilateral . Line 
intersects the lines , 
, at 
, 
, 
, respectively. If 
, show that 
.
whose largest interior angle is 
, lines 
and 
intersect at point 
, while lines 
and 
intersect at point 
. A point 
is taken in the interior of quadrilateral for which 
. 
is the circumcenter of quadrilateral . Line 
intersects the lines , 
, at 
, 
, 
, respectively. If 
, show that 
.
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is cyclic.
be the second intersection of 
with the circle.
.
and 
.
and 
,
.
.
,
is equal to 
,
is the polar of 
,
is the external bisector of 
,
is its internal bisector.
is harmonic.Suppose that 
.
is harmonic,which gives
is harmonic.Since 
is the internal bisector of 
,
is its external bisector.
and we are done.
are not necessary for the proof.
.
passes through 
and by Brocard we have 
.
hence 
is its internal angle bisector which implies 
.
are collinear. 
and 
hence 
.
as desired
