Problem #31
When Butter Sums It Up
The positive real numbers x_{0}, x_{1}, x_{2} ... x_{m} satisfy x_{0} = x_{m} and x_{i-1} + \frac{k}{x_{i-1}} = kx_{i} + \frac{1}{x_{i}}
Let x_{0}(k,m) be its maximum possible value of x_0 .
If (k=2 and m=59) , find the summation of such x_{0}(k,m) when you increase k by 1 and at the same time decrease m by 2 till both become equal.