such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
(use in a problem for junior) to practice. Can anyone share to me?
be a square with 
side length and with center 
.
is drawn where point 
.
Y by Adventure10
Two circles 
and 
, with centres at 
and 
respectively, touch at 
. The circle having 
as diameter intersects the circle at 
and the circle at 
. The points and both lie on the same side of the line . 
extended in both directions meets the circle at 
and meets the circle at 
. Prove that
(a)
(b) the line through perpendicular to bisects .
and 
, with centres at 
and 
respectively, touch at 
. The circle having 
as diameter intersects the circle at 
and the circle at 
. The points and both lie on the same side of the line . 
extended in both directions meets the circle at 
and meets the circle at 
. Prove that(a)
(b) the line through
perpendicular to bisects .
be the midpoint of 
.
and 
are similar whence 
and so 
.
so the same formula holds for 
and we are done.
be the point of intersection of that line through 
is tangent to both circles we have 
.
.
.
and hence 
which is a contradiction. Hence the claim.
