[/quote]
,
has circumcircle 
and incenter 
.
at 
respectively. Circumcircle of triangle 
intersects 
distinct from 
,
intersects 
distinct from 
.
points 
are concyclic.
is defined by 
and for all positive integers 
,
.
,
such that 
is a multiple of 
.
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 
.
and 
be two circles with respective centres 
and 
intersecting in two distinct points 
and 
is an obtuse angle. Let the circumcircle of 
intersect 
and 
.
intersect 
; let the line 
intersect 
.
are concyclic.
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
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 
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?
Y by Adventure10, Mango247
The opposite sides of a convex cluadrilateral are extended to intersect at two points. A line is drawn through each of
these points. These two lines divide the quadrilateral into four smaller quadrilaterals. If some pair of these quadrilaterals that don't share a common side, each has an inscribed circle, show that the original quadrilateral also has an inscribed
circle.
(I. Sharygin)
these points. These two lines divide the quadrilateral into four smaller quadrilaterals. If some pair of these quadrilaterals that don't share a common side, each has an inscribed circle, show that the original quadrilateral also has an inscribed
circle.
(I. Sharygin)
