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denotes the sum of the integers, 
denotes the sum of the square of the integers and 
denotes the sum of cubes of the integers(that is, if the three integers are 
, then 
, 
, 
). If 
and 
, find 
.
and 
that satisfy the system of equations:
and 
be the sets of integers and rationals respectively. into three non-empty subsets 
such that the sets 
are disjoint? into three non-empty subsets such that the sets are disjoint?
denotes the set 
, for 
and for 
.
be positive real numbers such that 
. Show that![\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]](http://latex.artofproblemsolving.com/e/a/1/ea1f56cc594862673040411c3ad322cb00cfe904.png)
If a problem has not been solved within 
days, then others are eligible to post a new probkem. Not only simple one-variable equations, but also systems of equations are allowed. The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one). Please indicate the domain of the solution to the equation (e.g., solve in 
, solve in 
).
and 
If each person remains there for 
minutes, what's the probability that all three people meet each other?
is an arithmetic sequence, 
, 
. Let the general term 
of the sequence 
be 
, and let 
be the sum of the 
terms of the sequence . Compare the size of with 
.
. Through point goes a straight line which intersects the circles at points and 
, so that is between and . The tangents to the circles, taken through points and , intersect at a point 
. Compare 
with 
.
. Through point goes a straight line which intersects the circles at points and , so that is between and . The tangents to the circles, taken through points and , intersect at a point . Compare with .
such that:
for all 
.
. Prove that
and 
are real numbers such that
and solve for the area of a hexagon in that 
square.
Prove that
Where 
hence 
is cyclic.
and 
,
,