[/quote]
,
:
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
so that the following relation holds for all 
.
and 
such that 
.
are positive isn't necessary.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
be a triangle, and let 
be a point on side 
.
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
.
and 
intersect 
and 
respectively.
to 
.
is extended beyond 
to 
so that 
.
,
be a cyclic quadrilateral. Let 
be the midpoint of the arc 
and 
and the lines 
and 
meet at 
.
and 
.
and 
respectively. Denote by 
the midpoint of the side 
and 
intersect at 
.
and 
have a common point lying on the side 
and 
be concentric circles, with 
)
and 
.
be a trapezoid with parallel sides 
.
and 
lie on the line segments 
and 
,
.
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points 
and 
are concyclic.
q
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?
,
and altitude 
.
.
meets 
meets 
be the orthogonal projection of 
meets 
.
.
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.
