Library / Austria competition problems /5 of 39

Austria competition problems, 2014

Algebra Difficulty 4.8 AIME Prove it Austria

Let $a$ , $b$ , $c$ and $d$ be real numbers with $a < b < c < d$ .
Sort the numbers $x = a b + c d$ , $y = b c + a d$ and $z = c a + b d$ in ascending order and prove the correctness of your result.

Solution

By the rearrangement inequality we have (writing $(\cdot, \cdot)$ for the scalar product of two vectors)
$\begin{aligned} y &= b c + a d = \langle (b, a), (c, d) \rangle < \langle (a, b), (c, d) \rangle = a c + b d = z \\ &= \langle (a, d), (c, b) \rangle < \langle (a, d), (b, c) \rangle = a b + c d = x. \end{aligned}$

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.