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.
be positive real numbers such that 
.
,
and 
.
be the intersection of 
and 
be the intersection of 
and 
is a cyclic quadrilateral prove that 
with only real zeros and real coefficients.
we have 
.
where 
denotes the Fibonacci sequence.**
is defined by the conditions
and 
for a set of indices 
.
satisfying the equation![\[
(I_p + A)^n = B^n + C^n.
\]](http://latex.artofproblemsolving.com/2/d/7/2d75d47203353160ccac356251659bd16a4cb5c8.png)
be a function that is at least twice differentiable on an open interval containing ![\( [0, 2\pi] \)](http://latex.artofproblemsolving.com/9/7/7/977adaa4f83dc8164975a3e78539f27cbb354adf.png)
.![\[
f(0) = f(2\pi) = f'(0) = f'(2\pi) = 0
\]](http://latex.artofproblemsolving.com/d/9/9/d998bdd53ffec1fa4c88688d7402ad25efa31112.png)
and![\[
f(x) + f''(x) \geq 0, \quad \forall x \in [0,2\pi].
\]](http://latex.artofproblemsolving.com/a/1/c/a1c24260aa02cd989630295d9de8741af659e958.png)
Prove that 
for all ![\( x \in [0,2\pi] \)](http://latex.artofproblemsolving.com/d/c/5/dc59f105fb3b9f6be175d2c035ba2ead4b9d984a.png)
.
,
and Moivre's formula.
.
,
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 
.
