List questions from sum-of-squares-method

Published on 25 Mar 2026 - 6:25

86

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Prove $\sum_{cyc} a(a-b)(a-2b) \ge 0$.Where $a,b,c \ge 0$

Published on 25 Mar 2026 - 6:12

103

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Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

Published on 04 Jun 2019 - 10:18

74

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Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2})\ge 4$

Published on 11 Jun 2019 - 4:38

772

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Prove $x+ y+ z= 3,\,x^{\,2}+ y^{\,2}+ z^{\,2}= 9\,\therefore\,y- x\leqq 2\sqrt{3}$ .

Published on 25 Mar 2026 - 1:20

86

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How to prove this inequality for $a,b,c>0$?

Published on 15 Jun 2019 - 20:54

265

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Given three positive numbers $a,\,b,\,c$ . Prove that $(\!abc+ a+ b+ c\!)^{3}\geqq 8\,abc(\!1+ a\!)(\!1+ b\!)(\!1+ c\!)$ .

Published on 23 Jun 2019 - 3:27

316

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Given two positive numbers $b,\,c$. Prove $\left ( \frac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \frac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ .

Published on 25 Mar 2026 - 6:13

89

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Given positive $a, b, c$, prove that $(a^2 + b^2 + c^2)^3 \ge (a^3 + b^3 + c^3)(ab + bc + ca)(a + b + c)$.

Published on 17 Jul 2019 - 15:12

126

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Solve inequality

Published on 20 Mar 2026 - 13:03

249

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Proving a reversed AM-HM inequality

Published on 30 Jul 2019 - 15:20

211

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How can we not use Muirhead's Inequality for proving the following inequality?

Published on 25 Mar 2026 - 6:12

60

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Inequality with a precondition

Published on 23 Aug 2019 - 16:01

96

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$a, b, c$ are positives such that $a + b + c = 1$. Determine the maximal value of $\sum\limits_{cyc}\frac1{a(b + c)} - \frac{a^2 + b^2 + c^2}{2abc}$.

Published on 10 Sep 2019 - 15:14

142

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show this inequality $\sum_{cyc}\frac{a^3}{a^2+ab+b^2}\ge\sqrt{\sum a^3}$

Published on 17 Sep 2019 - 15:53

List Question

Cyclic Olympiad Inequality

Prove $\sum_{cyc} a(a-b)(a-2b) \ge 0$.Where $a,b,c \ge 0$

Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2})\ge 4$

Prove $x+ y+ z= 3,\,x^{\,2}+ y^{\,2}+ z^{\,2}= 9\,\therefore\,y- x\leqq 2\sqrt{3}$ .

How to prove this inequality for $a,b,c>0$?

Given three positive numbers $a,\,b,\,c$ . Prove that $(\!abc+ a+ b+ c\!)^{3}\geqq 8\,abc(\!1+ a\!)(\!1+ b\!)(\!1+ c\!)$ .

Given two positive numbers $b,\,c$. Prove $\left ( \frac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \frac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ .

Given positive $a, b, c$, prove that $(a^2 + b^2 + c^2)^3 \ge (a^3 + b^3 + c^3)(ab + bc + ca)(a + b + c)$.

Solve inequality

Proving a reversed AM-HM inequality

How can we not use Muirhead's Inequality for proving the following inequality?

Inequality with a precondition

$a, b, c$ are positives such that $a + b + c = 1$. Determine the maximal value of $\sum\limits_{cyc}\frac1{a(b + c)} - \frac{a^2 + b^2 + c^2}{2abc}$.

show this inequality $\sum_{cyc}\frac{a^3}{a^2+ab+b^2}\ge\sqrt{\sum a^3}$

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