be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
and 
.
and 
.
,
,
Find max of 
(and prove) 
be positive integers and 
be an odd prime such that:![\[
a^p \equiv 1 \pmod{p^n}.
\]](http://latex.artofproblemsolving.com/3/4/8/34815b8211ded2f37b0653525dc4b436356e6eee.png)
Prove that:![\[
a \equiv 1 \pmod{p^{n-1}}.
\]](http://latex.artofproblemsolving.com/0/7/a/07a4ed2ab3c94f41028080f7789cfb6e2973955c.png)
Y by Eliot
Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most 
times the sum of their radii.
(A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.)
Note. Weaker results with replaced by some other 
may be awarded points depending on the value of 
Proposed by Morteza Saghafian
times the sum of their radii.(A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.)
Note. Weaker results with
replaced by some other 
may be awarded points depending on the value of 
Proposed by Morteza Saghafian
This post has been edited 2 times. Last edited by Gaussian_cyber, Nov 4, 2020, 6:34 PM
circles
lines tangent to all 
drawn and another circle with its center on the line 
try to have the actuall segment which the center moves on
then it is true for all the point on the segment 
?
