[/quote]
,
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
.
on the board?
and incenter 
.
at 
respectively. Circumcircle of triangle 
intersects 
,
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
Let 
be a triangle and let 
be points on the sides 
, 
, respectively, such that 
. Let 
be a point on side 
. Prove that if 
is perpendicular to 
then is the bisector of the angle 
, and conversely, if is the bisector of the angle then is perpendicular to .
be a triangle and let 
be points on the sides 
, 
, respectively, such that 
. Let 
be a point on side 
. Prove that if 
is perpendicular to 
then is the bisector of the angle 
, and conversely, if is the bisector of the angle then is perpendicular to .
