AoPS Academy
, 
, 
cut three common chords concurrent at 
. Consider two distinct circles 
, 
which are internally tangent to all 
. Determine, with proof, which of the following two statements is true. is the insimilicenter of and is the exsimilicenter of and .
be a parallelogram of center 
. Prove that for any point 
, there exist unique points 
and 
such that is the center of mass of 
.
+1 w with 
be given . On the extension of the segment 
beyond 
, a point 
is chosen arbitrarily. Let the line through and 
intersect the line 
at the point 
. Let 
be the intersection of the lines 
and 
. Prove that the line 
is tangent to the circumcircle of the triangle 
.
Thus it is enough to show that the equation![\[ 55 \cdot 3^n + 105(-1)^n = 27 \cdot 4^m + 48(-1)^m \]](http://latex.artofproblemsolving.com/1/3/6/136966708b4e9e3d41050010a1b3e8f615ff4387.png)
has no solutions over the integers. Assume otherwise for the sake of contradiction. One can easily check that there is no 
. Henceforth we assume 
. Then taken mod 27, we obtain![\[ 105 \equiv \pm 48 \pmod{27} \]](http://latex.artofproblemsolving.com/c/7/b/c7b73189faa5e7a21dbda293ae94a8036c498857.png)
which is false.
has no solutions for 
, because you have to prove that 
, not that 
and this gives 
and 
are two sequences of integers defined by
,
is 
is 
,and the roots 
and 
,we get 
.
where 
and 
are roots of the equation 
.
and 
we get 
.
.
we get 
.
if 
, so we get 
.
we get 
which is not true hence , sequence 
and 
shares no common numbers other than 