AoPS Academy
data values 
are all equal to 
. What is the value of ?
plot 
equally spaced points around a circle. Label one of them 
, and place a marker at . One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of 
distinct moves available; two from each point. Let 
count the number of ways to advance around the circle exactly twice, beginning and ending at , without repeating a move. Prove that 
for all 
.
be the real solution of the system of equations 
, 
, 
. The greatest possible value of 
can be written in the form 
, where 
and 
are relatively prime positive integers. Find 
.
.