[/quote]
,
has incentre 
and incircle touching 
at 
.
be the antipode of 
in the circumcircle of 
is chosen on the internal angle bisector of 
such that 
.
be the midpoint of arc 
,
be the midpoint of 
.
for which:
for all 
.
particles on a circle situated at the vertices of a regular 
be the smallest number of collisions after which all particles return to their original positions. Find 
![$Q\in \{-1,1,0\}[x]$](http://latex.artofproblemsolving.com/b/4/0/b404c8d6415aaf3d4442cc5381f582c1f8f8b996.png)
with 
.![$\exists P \in \mathbb Z[x]$](http://latex.artofproblemsolving.com/a/2/e/a2e67dd74df9971ca90d1a5d81c1c8e1f3982276.png)
palindrom with 
?
for which 
,
denote the integer part.
such that 
is divisible by 
,
is divisible by 
,
is divisible by 
.
and 
Prove that
chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
such that there exists a set 
of 
integers satisfying the following condition: for each 
,
such that 
is divisible by 
.
be the Fermat point of a 
.
,
,
are concurrent and the point of concurrence is 
,
be a family of functions from 
.
,
such that for all 
,![\[
y^2 \cdot f\left(\frac{g(x)}{y}\right) = h(xy)
\]](http://latex.artofproblemsolving.com/f/7/1/f7196164e0639695529b06788ea7cb1e50eddbe7.png)
and all 
,![\[
f\left(\frac{x}{f(x)}\right) = 1.
\]](http://latex.artofproblemsolving.com/1/5/b/15b363bf1754a4edd72423c3d0ea631173897beb.png)
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
and 
are given. Let 
be arbitrary points on 
.
.
of points in the plane is balanced if, for any two different points 
in 
in 
.
.
Y by Adventure10, Mango247, buddyram
The game of circulate is played with a deck of 
cards each with a number in 
such that there are 
cards with each number. First, 
piles numbered of cards each are dealt out face down. The player then flips over a card from pile 
, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game?
Brian Hamrick.
cards each with a number in 
such that there are 
cards with each number. First, 
piles numbered of cards each are dealt out face down. The player then flips over a card from pile 
, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game?Brian Hamrick.
s.t. each card in 
of the cards with numbers pointing to 
cards to start s.t. the number 
appears exactly 
times and the first card is to be drawn from any given pile, the result still holds. From this point on, we assume WLOG Hamster is female.
,
times unless 
times
,
cards with no self-contained subset, and assume that we can find a complete sequence of picks for any arrangement of cards according to the rules with piles and pile sizes identical to any proper subset of 
and a configuration with those unpicked, 
,
Can anybody please check it ?
is called shifu if it has the following properties:
,![$\text{Indeegree}[u] = \text{Outdegree}[u]$](http://latex.artofproblemsolving.com/a/5/4/a5443b543a73ef50983e20882f41038602758079.png)
to itself. (Though there might be multiple edges from 
,
.
is trivial. Now pick any vertex 
then pick two vertices 
such that 
and 
exist. and consider the graph 
obtained by taking 
.
,
.
from 
in 
.
into two disjoint sets 
such that for any 
,
or vice versa.
call a card appearing on 
useless cards. If on any rearrangement, if we operate on any useless card on a pile, we operate on that pile again, so taking away all the useless cards from each pile doesn't changes the outcome of the game. So we remove all the useless cards.
. So we create a directed graph 
,
edges is equal to the number of cards marked with the number ![$$\text{Indegree}[v_i] = \text{Total number of cards marked i on other piles} = k - m_i = \text{Number of cards in i th pile} = \text{Outdegree}[v_i]$$](http://latex.artofproblemsolving.com/a/a/8/aa8d619c249bc3384419ab60fe03231ff719fd89.png)
, so 
,
.
add a card marked 
to 
.
.
there exit card of number 
in 
by construction of graph). hence we will have our arrgement.
piles 
we should not have only number 
in these plies.
