Question 69
Mobile operator has $n$ clients and holds the following advertising campaign. In the beginning it deposits $1$ € on account of each client. When two persons which have $a$ € and $b$ € in their accounts communicate by a phone the operator makes both accounts equal to $(a + b)$ €. It happens that after $h(m)$ phone calls all $n$ clients' accounts become equal $m$ €. Prove that $h(m) \le \frac{1}{2} n \log_2 m$.