Problem

Source: China Girls Math Olympiad 2008, Problem 6

Tags: algebra unsolved, algebra



Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 - 7x_1)x_1^7 = 8$ and \[ x_{k + 1}x_{k - 1} - x_k^2 = \frac {x_{k - 1}^8 - x_k^8}{x_k^7x_{k - 1}^7} \text{ for }k = 2,3,\ldots \] Determine real number $ a$ such that if $ x_1 > a$, then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$, then the sequence is not monotonic.