[/quote]
,
is the diameter of circle 
,
and 
intersect at point 
,
intersects at point 
,
intersects 
.
lies on 
. Prove that points 
,
is incenter of triangle 
and 
is a line tangent to the incircle. Let 
be another line such that intersects 
respectively at 
.
to the incircle other than 
,
.
are similarly defined. Prove that 
are concurrent.
,
and 
.
.
and 
is right.
is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer 
on the board with 
,
.
is called 
if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
.
is tiled in two ways - only with dominoes and only with squares 
.
,
,
points 
and 
lie on the segments 
and 
,
is a parallelogram. The circle with center the midpoint 
intersect for the second time at the point 
.
and 
intersect in a point.
and 
Show that
Where 
Show that
Where 
and 
is closest to which integer?
Y by Sedro, Unicode_Master03B8, clarkculus, AmethystC, megarnie, Mogmog8
The 9 members of a baseball team went to an ice-cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let 
be the number of different assignments of flavors to players that meet these conditions. Find the remainder when is divided by 
be the number of different assignments of flavors to players that meet these conditions. Find the remainder when is divided by 
as well.
and 
.
to choose the people for chocolate and strawberry.
.
.
.
denote the number of people who chose each of the three flavors, where 
and 
.
.
,
was any smaller there would be no cases:
for symmetry reasons.
.
isn't assigned to anybody is 
.
aren't assigned to anybody is 
.
flavors with the same number of people liking each of them.
.
instead of 
to get an answer of 
so we get 016 as our final answer for this question.
vanilla, 
strawberry. We do casework on 
and 
:
ways to choose which members of the team had strawberry ice cream, then 
ways to choose which of the remaining members had vanilla ice cream, then the rest of the members must have had chocolate. Hence 
assignments for this case.
assignments.
assignments.
.
,
,
.![\[
(a, b, c)=(6, 2, 1), (5, 3, 1), (4, 3, 2).
\]](http://latex.artofproblemsolving.com/3/d/c/3dc5fe9187850dbe089cf6600b362994ae7d75aa.png)
and from a little easy calculation, we conclude that 
