1990 APMO Problems/Problem 2

We have to prove

Let, \begin{align*} S_k & = \sum_{i=1}^{n\choose k}t_i \\ S_{n-k} & = \left(\prod_{m=1}^{n}a_m \right)\left(\sum_{i=1}^{n\choose k}\dfrac{1}{t_i}\right) \end{align*} Thus, \begin{align*} S_k S_{n-k} &=\left(\sum_{i=1}^{n\choose k}t_i\right)\left(\prod_{m=1}^{n}a_m \right)\left(\sum_{i=1}^{n\choose k}\dfrac{1}{t_i}\right)\\ &=\left(\prod_{m=1}^{n}a_m \right)\left[\sum_{i=1}^{n\choose k}1+{\sum_{i=1}^{n\choose k}\sum_{j=1}^{n\choose k}}_{i \neq j}\dfrac{t_i}{t_j}\right]\\ &=\left(\prod_{m=1}^{n}a_m \right)\left[{n \choose k} + \sum_{i,j}\left(\dfrac{t_i}{t_j}+\dfrac{t_j}{t_i}\right)\right]\\ & \geq \left(\prod_{m=1}^{n}a_m \right)\left[{n \choose k}+2\cdot\dfrac{{n \choose k}^2-{n \choose k}}{2}\right]\\ &= {n\choose k}^2 \prod_{m=1}^{n}a_m \square \end{align*}

In the fourth line, we used the AM-GM inequality and used the fact that there are terms in the sum of third line.