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China Southeastern Mathematical Olympiad

Algebra Difficulty 5.7 AIME, harder Prove it China

Let $n \ge 3$ be integer. Suppose that $\alpha, \beta, \gamma \in (0, 1)$ and $a_k, b_k, c_k \ge 0$ $(k = 1, 2, \dots, n)$ satisfy $\sum_{k=1}^n (k+\alpha)a_k \le \alpha$ , $\sum_{k=1}^n (k+\beta)b_k \le \beta$ and $\sum_{k=1}^n (k+\gamma)c_k \le \gamma$ . Find the minimum of $\lambda$ such that $\sum_{k=1}^n (k+\lambda)a_k b_k c_k \le \lambda$ .

Solution

Let $a_1 = \frac{\alpha}{1+\alpha}$ , $b_1 = \frac{\beta}{1+\beta}$ , $c_1 = \frac{\gamma}{1+\gamma}$ , $a_i, b_i, c_i = 0$ $(i = 2, 3, \dots, n)$ . We see that all conditions are satisfied. So, we must have
$(1 + \lambda) \frac{\alpha}{1 + \alpha} \cdot \frac{\beta}{1 + \beta} \cdot \frac{\gamma}{1 + \gamma} \le \lambda,$
that is,
$\lambda \ge \frac{\alpha\beta\gamma}{(1+\alpha)(1+\beta)(1+\gamma) - \alpha\beta\gamma}.$

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