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$a,b,c$ are positive reals such that $ab+bc+ac=2022$, Prove that $$\frac{\sum_{cyc}\sqrt{a^2+2022}}{\sum_{cyc}\sqrt{ab}} \ge 2$$

I tried Holder but it doesn't seem to work. Also I don't know wether that's true but I think if we replace 2022 with $ab+bc+ac$ then the inequality is homogeneous and hence we can assume wlog that $\sum_{cyc}\sqrt{a^2+2022}$ is equal to whatever we want ? Am I right?

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  • $\begingroup$ Each term of the numerator must be at least as large as $\sqrt{2022}$, so "whatever we want" is a bit too strong of a phrase. $\endgroup$
    – dshin
    Commented Apr 2, 2022 at 16:50
  • $\begingroup$ I thought that replacing $2022$ with $ab+bc+ac$ would get rid of the costraint $ab+bc+ac=2022$ $\endgroup$
    – PNT
    Commented Apr 2, 2022 at 17:42
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    $\begingroup$ I see. Yes, that is justified. I might just reword your question slightly - it's the substituted numerator that can be assumed to equal anything. $\endgroup$
    – dshin
    Commented Apr 2, 2022 at 19:17

2 Answers 2

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As for the problem itself: note that $$ \sqrt{a^2+2022}=\sqrt{a^2+ab+ac+bc}=\sqrt{(a+b)(a+c)}\ge a+\sqrt{bc}~\text{(why?)}. $$ Can you continue now?

For the second question the answer is essentially yes: if you have a homogeneous expression such as $\sum_{cyc}\sqrt{a^2+ab+ac+bc}/\sum_{cyc}\sqrt{ab}$, you can always multiply $a$, $b$ and $c$ by some positive scalar $\lambda$ (here we assume that $a,b,c>0$) and then $\sum_{cyc}\sqrt{a^2+ab+ac+bc}$ will be also multiplied by $\lambda$. Hence, you can assume that $\sum_{cyc}\sqrt{a^2+ab+ac+bc}=L$ for any $L>0$.

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This is not a proof of the inequality, but simply an answer to your question.

Indeed, you can replace the $2022$ in the numerator with $ab+bc+ca$ to arrive at an unconstrained inequality that must be true if the original constrained inequality is true. To see this, make the substitution and assume for sake of contradiction that the substituted unconstrained inequality is false for some choice of $a, b, c$. Because the substituted unconstrained inequality is homogeneous, you can scale each $a, b, c$ by an appropriate constant to cause the constraint to become satisfied. But those scaled values then imply that the original constrained inequality is false, a contradiction. Thus, the substitution is justified.

After performing the substitution, the inequality is indeed homogeneous, and so the variables can be scaled arbitrarily without changing the value of the LHS.

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