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 
,
,
,
)
and 
,
.
and 
that satisfy the system of equations:
and 
be the sets of integers and rationals respectively.
such that the sets 
are disjoint?
denotes the set 
,
and for 
.
be positive real numbers such 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)
such that:
for all 
.
. Prove that
and 
are real numbers such that
If a problem has not been solved within 
days, then others are eligible to post a new probkem.
,
)
and 
If each person remains there for 
minutes, what's the probability that all three people meet each other?
and solve for the area of a hexagon in that 
square.
is an arithmetic sequence, 
,
.
of the sequence 
be 
,
be the sum of the 
terms of the sequence 
.
Prove that
Where 
Y by
Let (P, P^*), (Q, Q^*) be two pairs of isogonal conjugates wrt \triangle ABC, and \triangle P_AP_BP_C, \triangle P^*_AP^*_BP^*_C, \triangle Q_AQ_BQ_C, \triangle Q^*_AQ^*_BQ^*_C be circlecevian triangles of P,P^*,Q,Q^*.
The three lines A(P_AQ_A\cap P^*_AQ^*_A), B(P_BQ_B\cap P^*_BQ^*_B), C(P_CQ_C\cap P^*_CQ^*_C) are concurrent.
The Newton lines of quadrangles P_AQ_AP^*_AQ^*_A, P_BQ_BP^*_BQ^*_B, P_CQ_CP^*_CQ^*_C are sides of the Ceva triangle of the trilinear polar of the Newton line of the quadrangle PQP^*Q^*.
The Newton line of the quadrangle PQP^*Q^* and the line passing through the Vu circlecevian point of (P, Q) and PQ^*\cap P^*Q are parallel .
The three lines A(P_AQ_A\cap P^*_AQ^*_A), B(P_BQ_B\cap P^*_BQ^*_B), C(P_CQ_C\cap P^*_CQ^*_C) are concurrent.
The Newton lines of quadrangles P_AQ_AP^*_AQ^*_A, P_BQ_BP^*_BQ^*_B, P_CQ_CP^*_CQ^*_C are sides of the Ceva triangle of the trilinear polar of the Newton line of the quadrangle PQP^*Q^*.
The Newton line of the quadrangle PQP^*Q^* and the line passing through the Vu circlecevian point of (P, Q) and PQ^*\cap P^*Q are parallel .
