[/quote]
,
distinct positive integers and then writes in blue the 
sums of each two red numbers. At most how many of the blue numbers can be prime?
be a cyclic quadrilateral with circumcenter 
,
is not a diameter of its circumcircle. The lines 
,
is acute and let 
be its orthocenter. The points 
on the lines 
and 
.
,
.
be a primes 
and 
.
satisfying 
,
be the positive number satisfied 
Find the minimum of 
be positive integers such that no number divides some other number. If 
,
.
be real numbers such that 
.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
.
mutually non-attacking rooks placed on a grid 
.
Y by Tumon2001, Adventure10, and 1 other user
Let 
, 
and 
be three points on a line (in this order).
For each circle
through the points and , let 
be one point of intersection of the perpendicular bisector of 
with the circle . Further, let 
be the second point of intersection of the line 
with .
Show that for each circle, the ratio of lengths 
is the same.
, 
and 
be three points on a line (in this order).For each circle
through the points and , let 
be one point of intersection of the perpendicular bisector of 
with the circle . Further, let 
be the second point of intersection of the line 
with .Show that for each circle
, the ratio of lengths 
is the same.
is tangent to 
and 
(
and from this also 
) So,by similarity of 
, 
,we get 
and 
As 
.Therefore 
Hence the ratio is fixed.
and 
be two circles passing through 
and 
be the perpendicular bisector of 
.
= 
and 
.
= 
and 
,
= 
and let 
= 
.
and let 
= 
.
and 
.
is cyclic.
+ 
= 
°
= 
°
is also cyclic.
= 
is a diameter of 
passes through 
= 
(angle bisector theorem) = a constant.
