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sqing

#1 h Mar 28, 2025, 2:30 PM

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Let $a,b>0$ and $a^2+b^2=\frac{5}{2}.$ Prove that $2a + 2b + \frac{1}{a} + \frac{1}{b} +\frac{ab}{\sqrt 2}\geq 5\sqrt 2$ $a + b +\frac{2}{a} + \frac{2}{b} + ab\geq \frac{5}{4} + \frac{13}{\sqrt 5}$ $a + b +\frac{2}{a} + \frac{2}{b} + \frac{ab}{\sqrt 2}\geq \frac{5}{4\sqrt 2} + \frac{13}{\sqrt 5}$

This post has been edited 1 time. Last edited by sqing, Mar 28, 2025, 2:37 PM

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sqing

#2 h Mar 30, 2025, 12:33 PM

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Let $a,b>0$ and $ab+a+b=\frac{5}{2}.$ Prove that
$\frac{1}{a} + \frac{1}{b} \geq \frac{2(2+\sqrt{14})}{5}$ Let $a,b>0$ and $a^2+b^2+ab=\frac{5}{2}.$ Prove that
$2a + 2b + \frac{1}{a} + \frac{1}{b} +\frac{ab}{\sqrt 2}\geq 16\sqrt{ \frac{2}{5}}+ \frac{5}{6\sqrt 2}$ Let $a,b>0$ and $a^2+b^2+a+b=\frac{5}{2}.$ Prove that
$a + b +\frac{2}{a} + \frac{2}{b} + ab\geq \frac{47+42\sqrt 6 }{20}$ Let $a,b>0$ and $a^2+b^2+ab+a+b=\frac{5}{2}.$ Prove that
$a + b +\frac{1}{a} + \frac{1}{b} + 2ab \geq \frac{101+23\sqrt{34} }{45}$

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