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different positive integers , less than 
, there exist some two which are NOT relative prime.
satisfy,
Where {
} is a permutation of . Prove that 
be real numbers . Prove that
![$$-\frac{1}{18}\leq \frac{ab+a+b+3}{(a^2+3)(b^2+3)}\leq \frac{2+\sqrt[3] 2+\sqrt[3] {4}}{12} $$](http://latex.artofproblemsolving.com/9/9/9/9997397eb38cd4bceec2ff8c85951c78e2797d0f.png)
such that the equality 
holds for all pairs of real numbers 
.
is inscribed in a circle. The tangents to the circle through 
and 
intersect at a point 
, such that this point does not lie on 
, and such that 
. Prove that the line passes through the midpoint of 
.
is inscribed in a circle. The tangents to the circle through and intersect at a point , such that this point does not lie on , and such that . Prove that the line passes through the midpoint of .
be second intersection point of line 
and circle. Then
so 
.
is intersection point of
and circle. Now, 
, so
is a trapezoid.
is a polar of point 
wrt circle so lines 
and 
and
(remember and 
) is also passing through center
is a trapezoid), so it's passing through and through interection point of diagonals and
. So intersection point of lines 
and is midpoint of. The end
is an obvious solution. Now, let 
.
or 
.
.
