,
is the Gergon point and the incircle 
touch 
,
,
at 
,
,
respectively.
,
,
be 
, 
, 
respectively.
and then we get the circle 
and then the circle 
and then the circle 
and primes 
such that 
(there is at least 1 zero) can't be a perfect square.
such that![\[f((x\oplus f(y))+y)=(f(x)\oplus y)+y\]](http://latex.artofproblemsolving.com/d/3/5/d355f091797570bdc6c023eedb30e8ee9deac168.png)
Note: 
denotes the bitwise XOR operation. For example, 
.
Y by
There are seven rays emanating from a point 
on a plane, such that the angle between the two consecutive rays is 
. A point 
is located on the first ray. The projection of onto the second ray is denoted as 
. Similarly, the projection of onto the third ray is 
, and this process continues until the projection of 
onto the seventh ray is 
. Find the ratio 
.
Proposed by Giorgi Arabidze, Georgia
on a plane, such that the angle between the two consecutive rays is 
. A point 
is located on the first ray. The projection of onto the second ray is denoted as 
. Similarly, the projection of onto the third ray is 
, and this process continues until the projection of 
onto the seventh ray is 
. Find the ratio 
.Proposed by Giorgi Arabidze, Georgia
for all 
,![\[
\frac{AA_7}{AA_1}=\prod_{1\le i\le 6} \frac{AA_{i+1}}{AA_i} = \left(\frac{\sqrt3}{2} \right)^6
\]](http://latex.artofproblemsolving.com/e/8/b/e8bc93a541fb4b6ec90ee1d18575c5a731409d61.png)
which is equal to 
.
