AoPS Academy
be a cyclic quadrilateral with circumcentre 
. The diagonals 
and 
meet at point 
. Suppose the four incentres of triangles 
lie on a circle with centre 
. Prove that 
are collinear.
such that, for all positive integers 
and 
,
be a positive integer. We say that is bao-good if there exist integers 
such that:
for which![\[
a \mid s^b - l.
\]](http://latex.artofproblemsolving.com/5/d/9/5d9db7678c54d1c6a909b4aca39a329cadd71e2c.png)
2. For every proper divisor 
of (that is, 
and 
), there exists a positive integer such that![\[
a' \mid s^b - l.
\]](http://latex.artofproblemsolving.com/4/b/2/4b26b9bd3da8958287703c3f575bf5d5da7fd477.png)
Determine all bao-good positive integers .
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point .
on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point .
be the common point of the two circles. Let 
be one of the common external tangents. The circle we are searching for is the inverse of the line w.r.t. the circle 
. We obtain one more circle if we use the other external common tangent.
