List questions from sum-of-squares-method

Published on 23 Mar 2026 - 10:26

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If $2(bc^2+ca^2+ab^2) = b^2c+c^2a+a^2b+3abc, $ then prove that triangle $ABC$ is equilateral

Published on 22 Mar 2026 - 23:22

252

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Geometric (Trigonometric) inequality $\frac{(a+b+c)^3}{3abc}\leq1+\frac{4R}{r}$

Published on 23 Mar 2026 - 0:50

203

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Inequality with a+b+c=1

Published on 23 Mar 2026 - 2:56

350

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Proof that $\frac{a}{a+3b+3c}+\frac{b}{b+3a+3c}+\frac{c}{c+3a+3b} \ge \frac{3}{7}$ for all $a,b,c > 0$

Published on 25 Mar 2026 - 9:29

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Polynomial equal to sum of squares of polynomials

Published on 25 Mar 2026 - 9:27

269

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A cyclic, non-symmetric inequality

Published on 23 Mar 2026 - 6:02

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Proving a well-known inequality using S.O.S

Published on 23 Mar 2026 - 4:52

416

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Two inequalities involving the rearrangement inequality

Published on 25 Mar 2026 - 15:47

185

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For $x,y,z>0$. Minimize $P=\frac{1}{4x^{2}-yz+2}+\frac{1}{4y^{2}-zx+2}+\frac{1}{4z^{2}-xy+2}$

Published on 23 Mar 2026 - 0:03

57

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A Problem Involving an Inequality

Published on 22 Mar 2026 - 19:44

203

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For $2x+2y-3z\geq0$ and similar prove that $\sum\limits_{cyc}(7z-3x-3y)(x-y)^2\geq0$

Published on 23 Mar 2026 - 9:49

145

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Maximize $P=\frac{1}{\sqrt{1+x^{2}}}+\frac{1}{\sqrt{1+y^{2}}}+\frac{1}{\sqrt{1+z^{2}}}$

Published on 23 Mar 2026 - 12:39

81

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Minimize $S=\dfrac{x}{y^2}+\dfrac{y}{z^2}+\dfrac{z}{x^2}$ when $x+y+z=xyz$.

Published on 25 Mar 2026 - 17:21

148

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Let $a,b,c$ be positive numbers. Prove that $a^2b+b^2c+c^2a\leq a^3+b^3+c^3$

Published on 25 Mar 2026 - 17:19

List Question

Prove the inequality $(x+y+z)^2\ge 4(x^2y^2+y^2z^2+z^2x^2)$

If $2(bc^2+ca^2+ab^2) = b^2c+c^2a+a^2b+3abc, $ then prove that triangle $ABC$ is equilateral

Geometric (Trigonometric) inequality $\frac{(a+b+c)^3}{3abc}\leq1+\frac{4R}{r}$

Inequality with a+b+c=1

Proof that $\frac{a}{a+3b+3c}+\frac{b}{b+3a+3c}+\frac{c}{c+3a+3b} \ge \frac{3}{7}$ for all $a,b,c > 0$

Polynomial equal to sum of squares of polynomials

A cyclic, non-symmetric inequality

Proving a well-known inequality using S.O.S

Two inequalities involving the rearrangement inequality

For $x,y,z>0$. Minimize $P=\frac{1}{4x^{2}-yz+2}+\frac{1}{4y^{2}-zx+2}+\frac{1}{4z^{2}-xy+2}$

A Problem Involving an Inequality

For $2x+2y-3z\geq0$ and similar prove that $\sum\limits_{cyc}(7z-3x-3y)(x-y)^2\geq0$

Maximize $P=\frac{1}{\sqrt{1+x^{2}}}+\frac{1}{\sqrt{1+y^{2}}}+\frac{1}{\sqrt{1+z^{2}}}$

Minimize $S=\dfrac{x}{y^2}+\dfrac{y}{z^2}+\dfrac{z}{x^2}$ when $x+y+z=xyz$.

Let $a,b,c$ be positive numbers. Prove that $a^2b+b^2c+c^2a\leq a^3+b^3+c^3$

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