List questions from cauchy-schwarz-inequality

Published on 25 Mar 2026 - 19:06

113

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If $x,y,z>0$, prove that $\frac{x+y+z}{3\sqrt{3}} \geq \frac{yz+zx+xy}{\sum \limits_{cyc} \sqrt{x^2+xy+y^2}}$

Published on 29 Mar 2026 - 7:40

235

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An inequality concerning Lagrange's identity

Published on 20 Feb 2017 - 15:15

211

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Prove that $\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{9a^2+b^2}}+\frac{2ab}{\sqrt{a^2+b^2}\times \sqrt{9a^2+b^2} }\leq \frac32.$

Published on 29 Mar 2026 - 5:43

153

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Prove the inequality, fractions.

Published on 29 Mar 2026 - 4:09

100

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Minimize $P=5\left(x^2+y^2\right)+2z^2$

Published on 29 Mar 2026 - 5:49

119

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Triangular inequality extended

Published on 24 Feb 2017 - 23:27

106

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Prove that $n \geq \sqrt{n-1} + \sqrt{n}$ for $n \geq 4$

Published on 27 Feb 2017 - 4:55

208

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Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$

Published on 29 Mar 2026 - 5:43

121

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Finding maximum of $x+y$

Published on 25 Mar 2026 - 19:01

130

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Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$

Published on 29 Mar 2026 - 11:02

126

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Hint needed to show that $a\cos t+b\sin t\leq \sqrt{a^2+b^2}$ for $t\in [0,2\pi)$

Published on 05 Mar 2017 - 4:04

233

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Proving: $\left(a+\frac1a\right)^2+\left(b+\frac1b\right)^2\ge\frac{25}2$

Published on 26 Mar 2026 - 2:56

111

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For $a,b,c>0$ prove that $\frac{a}{a+\sqrt{(a+b)(a+c)}} + \frac{b}{b+\sqrt{(b+a)(b+c)}}+ \frac{c}{c+\sqrt{(c+b)(c+a)}} \leq 1$

Published on 07 Mar 2017 - 14:35

271

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For $a,b,c$. Prove that $\frac{a^2}{a+\sqrt[3]{bc}}+\frac{b^2}{b+\sqrt[3]{ca}}+\frac{c^2}{c+\sqrt[3]{ab}}\ge\frac{3}{2}$

Published on 29 Mar 2026 - 10:49

List Question

Minimize $\big(3+2a^2\big)\big(3+2b^2\big)\big(3+2c^2\big)$ if $a+b+c=3$ and $(a,b,c) > 0$

If $x,y,z>0$, prove that $\frac{x+y+z}{3\sqrt{3}} \geq \frac{yz+zx+xy}{\sum \limits_{cyc} \sqrt{x^2+xy+y^2}}$

An inequality concerning Lagrange's identity

Prove that $\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{9a^2+b^2}}+\frac{2ab}{\sqrt{a^2+b^2}\times \sqrt{9a^2+b^2} }\leq \frac32.$

Prove the inequality, fractions.

Minimize $P=5\left(x^2+y^2\right)+2z^2$

Triangular inequality extended

Prove that $n \geq \sqrt{n-1} + \sqrt{n}$ for $n \geq 4$

Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$

Finding maximum of $x+y$

Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$

Hint needed to show that $a\cos t+b\sin t\leq \sqrt{a^2+b^2}$ for $t\in [0,2\pi)$

Proving: $\left(a+\frac1a\right)^2+\left(b+\frac1b\right)^2\ge\frac{25}2$

For $a,b,c>0$ prove that $\frac{a}{a+\sqrt{(a+b)(a+c)}} + \frac{b}{b+\sqrt{(b+a)(b+c)}}+ \frac{c}{c+\sqrt{(c+b)(c+a)}} \leq 1$

For $a,b,c$. Prove that $\frac{a^2}{a+\sqrt[3]{bc}}+\frac{b^2}{b+\sqrt[3]{ca}}+\frac{c^2}{c+\sqrt[3]{ab}}\ge\frac{3}{2}$

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