[/quote]
,
be a root of 
.
satisfy ![\[a+b+c+d=0,\qquad \frac1a+\frac1b+\frac1c+\frac1d=0.\]](http://latex.artofproblemsolving.com/9/5/8/95849c32951bb80891e85d4b04da6034d328818e.png)
How many equalities ![\[ab=cd,\qquad ac=bd,\qquad ad=bc\]](http://latex.artofproblemsolving.com/2/9/7/29730d243048a530d2078d80daf0ca2522272554.png)
can simultaneously hold? Determine all possibilities.
be an acute triangle, 
its orthocenter, 
circumcenter and 
midpoint of 
Denote 
the intersection of line 
and the circumcircle 
intersection of line 
and 
Finally, let 
be the intersection of line 
and the circumcircle of 
Show that 
of a set of 21 distinct real numbers such that ![\[20|x-y|<(x+1)(y+1)?\]](http://latex.artofproblemsolving.com/3/0/e/30e7612bb44e1c889ec778ce928d258ea0c381d0.png)
such that 
cells of 
square grid can be colored in a way where no two dyed squares share a point and there are exactly two dyed squares in every column and every row.
(mod 4), determine the number of ordered integer triplets 
such that
be a sequence of positive numbers such that ![\[a_{n+1}=a_n+\frac{1}{a_n}\]](http://latex.artofproblemsolving.com/0/9/e/09ee35216612db492e63b57252ca6080c2c28f41.png)
for all positive integers 
Determine the greatest integer 
such that exactly 
)
be a positive integer and 
its greatest prime divisor. For each non-empty subset of divisors of 
write the sum of its elements on the board. Assume that more than 
are written and any of them occurs at most once. Show that all numbers on the board are distinct.
of real numbers, let 
denote the sum of the elements of 
of 
given by three of these points such that 
with center at point 
inscribed in an triangle 
)
,
and 
at points 
,
and 
,
intersect secondary at point 
The lines 
and 
intersect at point 
and intersects the line 
and 
,
and 
,
and 
intersect at the point 
.
then 
.
and let
be 
subsets of 
satisfying the following conditions:
elements of 
;
for all 
.
(with 
)
elements in common.
with integer coefficients satisfy the condition: there exists a integer 
so that for all integers 
,
is a perfect square, then there exists 
and a polynomial 
with integer coefficients with 
?
Y by
Can someone help me with this problem? Let 
. We call a distribution the act of distributing the integers from 
to
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
We say that
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
, the number of chicken-fox-viper distributions?
. We call a distribution the act of distributing the integers from 
to
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.We say that
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
, the number of chicken-fox-viper distributions?
?
,
are the draws of players 
and 
.
,
.
would satisfy 
with equal probability 
,
.
,
and player 
has 
"
