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Problem 882

AMC 12 late, AIME early

Algebra Difficulty 4.6 Prove it All-Soviet-Union Mathematical Olympiad · Soviet Union

Given four positive real numbers $a$ , $b$ , $c$ , $d$ such that $abcd = 1$ , prove that
$a^2 + b^2 + c^2 + d^2 + ab + ac + ad + bc + bd + cd \geq 10.$

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:
Applying the arithmetic/geometric mean result to the 10 numbers gives the result immediately.

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