List questions from a-m-g-m-inequality

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Proving that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1\implies (a+1)(b+1)(c+1)\geq 64$ where $a,b,c>0$.

Published on 30 Sep 2020 - 4:53

90

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If $x,y,z\in(1,3)$ and $xy+yz+zx=26$ then prove that $x+y+z>\frac{35}{4}$(Sweden 1971)

Published on 01 Oct 2020 - 10:50

127

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You are given the sides of a triangle $a,b,c$ Prove that: $a^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le 3abc$ (Sweden 1950)

Published on 01 Oct 2020 - 13:53

131

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A cyclic inequality of degree 10

Published on 26 Mar 2026 - 23:11

113

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prove $\sum_{cyc}\frac{a^2}{a+2b^2}\ge 1$

Published on 31 Mar 2026 - 19:44

656

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Maximum of $\sin(x)\cos(y) + \sin(y)\cos(z) + \sin(z)\cos(x)$

Published on 09 Oct 2020 - 18:41

121

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Minimum value of this trigonometric function

Published on 10 Oct 2020 - 9:46

137

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If $a+b+c=3$, then show $\sqrt{a^2 + ab +b^2}+ \sqrt{b^2 + bc +c^2}+\sqrt{c^2 + ac +a^2} \geq \sqrt{3}$

Published on 12 Oct 2020 - 10:57

71

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If $a, b>0$ and n is a positive integer, prove that: $(1+\frac{a}{b})^n+(1+\frac{b}{a})^n\ge 2^{n+1}$

Published on 14 Oct 2020 - 15:09

67

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Have I done my homogenization correctly for this question and if so how do I finish it off?

Published on 14 Oct 2020 - 15:23

151

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Proving a cyclic inequality

Published on 25 Mar 2026 - 23:17

352

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Max min with $xy^2+yz^2$

Published on 20 Oct 2020 - 12:13

99

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Prove that if $x, y, z$ are positive then $\frac{2}{1/x+1/y+1/z} \le \sqrt[3]{xyz}$

Published on 23 Oct 2020 - 1:32

88

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If $b^2+c^2+bc=3$ then $b+c\leq 2$

Published on 31 Mar 2026 - 21:34

69

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prove AM-GM inequality for $n=k-1$

Published on 23 Oct 2020 - 9:09

List Question

Proving that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1\implies (a+1)(b+1)(c+1)\geq 64$ where $a,b,c>0$.

If $x,y,z\in(1,3)$ and $xy+yz+zx=26$ then prove that $x+y+z>\frac{35}{4}$(Sweden 1971)

You are given the sides of a triangle $a,b,c$ Prove that: $a^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le 3abc$ (Sweden 1950)

A cyclic inequality of degree 10

prove $\sum_{cyc}\frac{a^2}{a+2b^2}\ge 1$

Maximum of $\sin(x)\cos(y) + \sin(y)\cos(z) + \sin(z)\cos(x)$

Minimum value of this trigonometric function

If $a+b+c=3$, then show $\sqrt{a^2 + ab +b^2}+ \sqrt{b^2 + bc +c^2}+\sqrt{c^2 + ac +a^2} \geq \sqrt{3}$

If $a, b>0$ and n is a positive integer, prove that: $(1+\frac{a}{b})^n+(1+\frac{b}{a})^n\ge 2^{n+1}$

Have I done my homogenization correctly for this question and if so how do I finish it off?

Proving a cyclic inequality

Max min with $xy^2+yz^2$

Prove that if $x, y, z$ are positive then $\frac{2}{1/x+1/y+1/z} \le \sqrt[3]{xyz}$

If $b^2+c^2+bc=3$ then $b+c\leq 2$

prove AM-GM inequality for $n=k-1$

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