[/quote]
,
be an isosceles triangle with 
.
be a point on 
.
be a point inside the triangle such that 
and![\[
CL \cdot BD = BL \cdot CD.
\]](http://latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.
satisfying 
and
for all 
.
Prove that
be a cyclic quadrilateral with 
being the midpoints of segments 
respectively. Suppose 
is the intersection of diagonals 
of quadrilateral 
Define 
to be the isogonal conjugate point of point 
Define 
similarly. Suppose 
intersects 
at a point 
Prove that: The Newton-Gauss line of quadrilateral 
Prove that
Prove that
be an odd prime number. How many 
of 
are there, the sum of whose elements is divisible by 
and a prime number 
.
is nonconstant, 
,
for all integers 
.
.
in order. Ana wins if she can determine the value of 
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through 
to 
such that 
.
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of 
,
it's circumcenter. Let 
and 
the intersection of the circumcircle of 
,
is the reflection of 
the midpoint of 
,
and the parallel through 
,
and the perpendicular through 
and 
a point on 
,
is the midpoint of 
.
lie on a line.
is inscribed in a circle 
.
— the circle inscribed in a segment 
circle of 
— the centers of circles 
and 
are parallel.
intersect at 
intersect at 
.
.
with the diagonals of 
and 
are tangent at a fixed point.
,
.
and 
,
is acute, and let 
and 
,
and 
and 
and 
Y by sami1618
Ali is hosting a large party. Together with his 
friends, 
people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).
Find all values of such that after some number of steps, the situation reaches a point where each person has exactly one apple.
friends, 
people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).Find all values of
such that after some number of steps, the situation reaches a point where each person has exactly one apple.
work inductively as it goes from 
to two copies 
which are apart, this decays into all ones.
turn into a 
component with 
and 
zeros before them. The 
decays the same way so we can finish inductively to get that it never is all ones. Thus this ends up becoming 
inch worms in different states which never have the same all 
time which gives the result.
for all non-negative integers 
.
steps, everyone will have exactly one apple. We proceed by induction on 
is trivial. For the induction step, assume that the result holds for 
.
apples, the 
people will have exactly 
are even.
,
works, then 
also works.
for a non-negative integer 
.
must also work. But by Claim 1, this is a contradiction, as desired.
