Given any $a,b,c \geq 1$, prove that:
$a^2 + b^2 + c^2 \geq 2a\sqrt{b-1} + 2b\sqrt{c-1} + 2c\sqrt{a-1}$
I tried using most of the popular inequalities and I didn't end up anywhere. Can anyone guide me through this problem?
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2$\begingroup$ Welcome to Mathematics SE. Take a tour. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. $\endgroup$– Another UserCommented Oct 16, 2022 at 14:18
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1 Answer
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$RHS^2 \leq (4a^2+4b^2+4c^2)(a+b+c-3)$, so it suffices to show $a+b+c -3 \leq \frac{1}{4}(a^2+b^2+c^2)$
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3$\begingroup$ Welcome to MSE. This question goes against our quality standard policy. Instead of posting an answer, you should encourage the person who posted the question to improve it. $\endgroup$ Commented Oct 16, 2022 at 14:32
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2$\begingroup$ Thank you! Your response really helped me understand the problem properly. $\endgroup$– David399Commented Oct 16, 2022 at 14:33
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1$\begingroup$ oh sorry, I should have known this better... I'll bear this in mind next time. $\endgroup$– LNTCommented Oct 16, 2022 at 14:34
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1$\begingroup$ @David399, you're welcome, guess we have to read carefully the policy next time :). $\endgroup$– LNTCommented Oct 16, 2022 at 14:41
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$\begingroup$ You're absolutely right! I will take care of the way I write my question next time. $\endgroup$– David399Commented Oct 16, 2022 at 14:47