balls, with 
red, 
be complex numbers such that [center]
for any 
with 
[/center]
and 
for any 
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
.
be a point on the arc 
(not containing 
and 
.
and 
intersect at point 
.
be the reflection of 
and 
intersect at point 
,
.
.
,
,
we have ![\[ 3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}. \]](http://latex.artofproblemsolving.com/a/9/8/a98b6c29802355c491016bcce58228a020423526.png)
be an integer. Find, with proof, all sequences 
of positive integers with the following three properties:
;
for all 
;
and 
(not necessarily distinct) for which 
,
such that 
.
boys and 
girls line up in a row. Let 
be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row 
we have 
.
satisfy 
?
be an integer. Find all positive real solutions to the following system of 
equations:
for which 
is a divisor of 
Y by
Let 
be a triangle and 
denote its inradius and ex-radii opposite to the vertices 
respectively. If 
and 
, then which of the following is/are true?
(A)
is obtuse
(B)
is acute
(C)
, where 
is semi perimeter
(D)
, where is semi perimeter
be a triangle and 
denote its inradius and ex-radii opposite to the vertices 
respectively. If 
and 
, then which of the following is/are true?(A)
is obtuse(B)
is acute(C)
, where 
is semi perimeter(D)
, where is semi perimeter
