Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Prove it Greece

Let $a$ , $b$ , $c$ , $d$ be positive real numbers such that
$a^2 + b^2 + c^2 + d^2 = 4.$
Prove that there are two of $a$ , $b$ , $c$ , $d$ with sum greater or equal to $2$ .

Solution

Without loss of generality, let $a \ge b \ge c \ge d$ and we will prove that $a+b \ge 2$ .
We have that $ab \ge c^2$ and $ab \ge d^2$ , so by adding them we have: $2ab \ge c^2 + d^2$ .
Therefore,
$(a+b)^2 = a^2 + b^2 + 2ab \ge a^2 + b^2 + c^2 + d^2 = 4,$
so $a+b \ge 2$ .

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