Se dan números reales positivos $a_1, \dots, a_k, b_1, \dots, b_k$. Sea $A = \sum_{i = 1}^k a_i, B = \sum_{i = 1}^k b_i$. Demuestra la desigualdad \[ \left( \sum_{i = 1}^k \frac{a_i b_i}{a_i B + b_i A} - 1 \right)^2 \ge \sum_{i = 1}^k \frac{a_i^2}{a_i B + b_i A} \cdot \sum_{i = 1}^k \frac{b_i^2}{a_i B + b_i A}. \]