Track / Stage 7 / 228 of 300 #1628 of 2000

Problem 1628

National olympiad second round; IMO P1/P4

Algebra Difficulty 7.5 Prove it

47. Let $a_{1}, a_{2}, \cdots, a_{n} ; b_{1}, b_{2}, \cdots, b_{n}$ be two sequences of positive numbers. Prove: $\left(\sum_{i \neq j} a_{i} b_{j}\right)^{2} \geqslant$ $\left(\sum_{i \neq j} a_{i} a_{j}\right)\left(\sum_{i \neq j} b_{i} b_{j}\right) \cdot(1998$ Yugoslav Mathematical Olympiad problem)

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

47. Let $A=\sum_{i=1}^{n} a_{i}, B=\sum_{i=1}^{n} b_{i}$ , the problem is equivalent to proving: $\left(A B-\sum_{i=1}^{n} a_{i} b_{i}\right)^{2} \geqslant\left(A^{2}-\right.$ $\left.\sum_{i=1}^{n} a_{i}^{2}\right)\left(B^{2}-\sum_{i=1}^{n} b_{i}^{2}\right)$

Construct a quadratic function: $f(x)=\left(A^{2}-\sum_{i=1}^{n} a_{i}^{2}\right) x^{2}+2\left(A B-\sum_{i=1}^{n} a_{i} b_{i}\right) x+\left(B^{2}-\sum_{i=1}^{n} b_{i}^{2}\right)$ .
We need to prove that its discriminant $\Delta \geqslant 0$ , now we just need to verify that there exists a real number $x_{0}$ such that $f\left(x_{0}\right) \leqslant 0$ , take
$x_{0}=-\frac{B}{A}, f\left(-\frac{B}{A}\right) \leqslant 0 \Leftrightarrow B^{2} \sum_{i=1}^{n} a_{i}^{2}-2 A B \sum_{i=1}^{n} a_{i} b_{i}+A^{2} \sum_{i=1}^{n} b_{i}^{2} \geqslant 0$

To prove (1), we just need to prove that its discriminant $\Delta_{1} \leqslant 0$ , which is obtained by the Cauchy-Schwarz inequality $\left(\sum_{i=1}^{n} a_{i} b_{i}\right)^{2} \leqslant$ $\sum_{i=1}^{n} a_{i}^{2} \sum_{i=1}^{n} b_{i}^{2}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.