be real numbers so that 
and 
Show that ![$a\in[0,\frac{4}{3}]$](http://latex.artofproblemsolving.com/0/e/c/0ecb933d7b008114469c3915dc2f8cbf43e362dc.png)
,![$b\in[0,\frac{2}{3}]$](http://latex.artofproblemsolving.com/8/4/1/841685820e03663b8ba4b65615e7ecf40fc31bcc.png)
,![$c\in[0,\frac{4}{9}]$](http://latex.artofproblemsolving.com/3/3/4/3349e77972554eb938563cef5aed9425367cd48d.png)
and
where 
is a prime
such that 
the given condition is false.
where 
is a natural number.What is the remainder when 
Y by
Let 
be the bisector of an acute-angled triangle 
. The circle circumscribed around the triangle 
intersects the straight line perpendicular to that passes through point 
, at point 
. Point 
is the center of the circumscribed circle of triangle . Prove that the points 
lie on the same line.
be the bisector of an acute-angled triangle 
. The circle circumscribed around the triangle 
intersects the straight line perpendicular to that passes through point 
, at point 
. Point 
is the center of the circumscribed circle of triangle . Prove that the points 
lie on the same line.
at 
and
Thus, 
,
are collinear as desired.
