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 .
+2 w
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 
.
. Then, 
if and only if 
. Since 
, we have 
, so if 
where 
are primes, then 
. Suppose that not all of the are equal. Then, 
can only appear in the prime factorization less than times, so if it appears 
times, then 
, so since 
for 
, this means that 
, which is a contradiction. Therefore, appears times in the prime factorization of 
, so no other prime can appear. Therefore, if and only if 
for some prime 
. The smallest such number at least 
is 
.
to get![\[
g(N) = \sum_{1\le i\le k}\frac{\alpha_i}{p_i},\quad\text{where}\quad N = \prod_{1\le i\le k}p_i^{\alpha_i}
\]](http://latex.artofproblemsolving.com/7/a/1/7a17a49f1cb88c8b74defff78650ed5cbb8f4c2a.png)
for primes 
. Now, 
. Let be as above. Multiplying both sides by 
where 
, we get 
. Hence 
. From here, 
for some prime . From here, we immediately conclude 
to be the smallest such number.
be a function such that
for every prime 
for every pair of positive integers 
such that 
