,![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
,
be all the positive integers smaller than 
such that
for all 
.
be the n-th prime number in increasing order for 
.
satisfying 
for all 
Y by Adventure10
The diagonals 
and 
of a cyclic quadrilateral meet at 
. The midpoints of the sides 
and 
are 
and 
respectively. Prove that the circles 
and 
have the same radius.
and 
of a cyclic quadrilateral meet at 
. The midpoints of the sides 
and 
are 
and 
respectively. Prove that the circles 
and 
have the same radius.
, so it's enough to prove that 
.
and 
are similar (in this order) , so their corresponding median 
, cut the same angle at 
.
in 
.
,
is cyclic. But the homothetie
take 
and 
to 
and 
so we're done.
