be a triangle with incentre 
and circumcircle 
.
be the midpoints of the arcs 
of 
respectively. Let 
be the point of 
.
and the midpoint 
of 
lie on a line.
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
lies on diagonal 
such that 
.
.
has 
,
,
.
.
Y by
From a point 
inside a square 
we raise a segment 
perpendicular to the plane of the square. Show that the projections of on the planes 
, 
, 
and 
are coplanar if and only if ![$O\in [AC]\cup [BD]$](//latex.artofproblemsolving.com/6/6/6/666df8f2afa87f15c79f9c96d9d1cea17a2a7af0.png)
.
inside a square 
we raise a segment 
perpendicular to the plane of the square. Show that the projections of on the planes 
, 
, 
and 
are coplanar if and only if ![$O\in [AC]\cup [BD]$](http://latex.artofproblemsolving.com/6/6/6/666df8f2afa87f15c79f9c96d9d1cea17a2a7af0.png)
.
