Competition problem — Algebra — Maths Olympiad Prep

Algebra Difficulty 5.1 AIME, harder Prove it Estonia

Does there exist a positive real number $C$ such that the inequality
$x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4 \le C(x_1x_2 + x_2x_3 + x_3x_4 + x_4x_1)$
holds for arbitrary positive real numbers $x_1, x_2, x_3, x_4$ ?

Solution

For simplicity, denote $A = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4$ and $B = x_1x_2 + x_2x_3 + x_3x_4 + x_4x_1$ .

Choose $x_1 = x_3 = u$ and $x_2 = x_4 = 1$ . Then $A \ge x_1x_3 = u^2$ and $B = 4u$ . Hence $\frac{A}{B} \ge \frac{u}{4}$ . As $u$ can be arbitrarily large, no constant $C$ such that $A \le CB$ exists.

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