Question 43
Fix a positive real number $c\gt 1$ and positive integer $n$. Initially, a blackboard contains the numbers $1, c, \ldots, c^{n-1}$. Every minute, Bob chooses two numbers $a, b$ on the board and replaces them with $c a + c^{2} b$. Prove that after $n-1$ minutes, the blackboard contains a single number no less than
$$
\left(\frac{c^{n / L}-1}{c^{1 / L}-1}\right)^{L}
$$
where $\phi=\frac{1+\sqrt{5}}{2}$ and $L=1+\log _{\phi}(c)$.
$$
\left(\frac{c^{n / L}-1}{c^{1 / L}-1}\right)^{L}
$$
where $\phi=\frac{1+\sqrt{5}}{2}$ and $L=1+\log _{\phi}(c)$.