Baltic Way 1990 · Problem 4

Algebra

Prove that, for any real $a_{1}, a_{2}, \ldots, a_{n}$ ,

\sum_{i, j=1}^{n} \frac{a_{i} a_{j}}{i+j-1} \geq 0

When you’re ready

Review material becomes available with the next Daily.

Review

Sequences and recurrences · Equations and inequalities

Solution

Solution: Consider the polynomial $P(x) = a_{1} + a_{2} x + \cdots + a_{n} x^{n-1}$ . Then $P^{2}(x) = \sum_{k, l=1}^{n} a_{k} a_{l} x^{k+l-2}$ and

\int_{0}^{1} P^{2}(x) d x = \sum_{k, l=1}^{n} \frac{a_{k} a_{l}}{k+l-1}.