and 
are in the coordinate plane such that 
.
denote the locus of all points 
in the coordinate plane satisfying 
,
be the midpoint of 
.
and 
are on 
and 
.
can be expressed in the form 
,
and 
are relatively prime positive integers. Find 
.
and 
Prove that 
Y by
Given the polynomial 
, where 
are real numbers. Suppose that 
has three distinct real roots and the polynomial 
has no real roots. Prove that
![\[
P(1) > 69^3.
\]](//latex.artofproblemsolving.com/2/a/4/2a49f9db6af78d720d5a657737625c011a1b554a.png)
, where 
are real numbers. Suppose that 
has three distinct real roots and the polynomial 
has no real roots. Prove that![\[
P(1) > 69^3.
\]](http://latex.artofproblemsolving.com/2/a/4/2a49f9db6af78d720d5a657737625c011a1b554a.png)
has no real roots, and 
takes all real values 
,
.
,
,
--- and so
