Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.6 AIME Prove it Soviet Union

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

Solution

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

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