[/quote]
,
be positive real numbers such that 
.
![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](http://latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers 
,
,
,
,
.
be a prime and suppose 
is prime. Prove that 
is prim
are positive real numbers such that 
prove that
girls on one side of a tennis table, and a queue of 
to 
grid. We color the squares with three colors: blue, red, and yellow. Two squares are defined as opposite if they share a vertex but not an edge. A valid coloring is a coloring such that no two squares that are red and blue are opposite each other. Count the number of valid colorings.
,
and 
.
be the intersection of 
and 
be the intersection of 
and 
is a cyclic quadrilateral prove that 
,
boys (with 
)
boys and for any 
boys, there is exactly 
be the greatest number of girls that can be chosen from the camp to form a group such that every boy is friend with at most 
.
such that 
is a perfect square.
,
is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?
,
.
Y by
In a convex quadrilateral 
, diagonals 
and 
are equal, and they intersect at 
. Perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
, and perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
. Prove that 
.
, diagonals 
and 
are equal, and they intersect at 
. Perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
, and perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
. Prove that 
.
