be a cyclic quadrilateral with circumcentre 
.
meet at point 
lie on a circle with centre 
.
are collinear.
such that, for all positive integers 
and 
,
be a positive integer. We say that 
such that:
for which![\[
a \mid s^b - l.
\]](http://latex.artofproblemsolving.com/5/d/9/5d9db7678c54d1c6a909b4aca39a329cadd71e2c.png)
2. For every proper divisor 
of 
and 
)![\[
a' \mid s^b - l.
\]](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)
Y by Davi-8191, Adventure10, Mango247
Let the circles 
and 
intersect in 
and 
. Tangent to circle at intersects in 
and tangent to circle at intersects in 
. Suppose that 
intersects and in 
and 
, respectively (assume that is between and ). The perpendicular to 
from intersects in point 
and perpendicular to 
from intersects in point 
(The points 
and lie on the same side of the line ). Prove that the points and are collinear.
Proposed by Mahdi Etesami Fard
and 
intersect in 
and 
. Tangent to circle at intersects in 
and tangent to circle at intersects in 
. Suppose that 
intersects and in 
and 
, respectively (assume that is between and ). The perpendicular to 
from intersects in point 
and perpendicular to 
from intersects in point 
(The points 
and lie on the same side of the line ). Prove that the points and are collinear.Proposed by Mahdi Etesami Fard
This post has been edited 1 time. Last edited by MRF2017, Sep 13, 2016, 5:10 AM
Reason: subject changed!
Reason: subject changed!
and let 
.
and 
to 
such that 
.
and 
and 
respectively, and 
we get 
.
.
.
.
.
.
is a cyclic quadrilateral with center 
.
.
.
will be collinear only if 
