query: Minimum value of $\,\left(a^3 + b^3 + c^3 -3abc\right)$
positive_passages.docid: 155493
positive_passages.text: Minimum value of the expression given below.
negative_passages.docid: 45775
negative_passages.text: In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$
negative_passages.docid: 13220
negative_passages.text: find the minimum value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$
negative_passages.docid: 26617
negative_passages.text: Minimum value of $ f(x,y,z)=\left(x+\frac{1}{y}\right)^2+\left(y+\frac{1}{z}\right)^2+\left(z+\frac{1}{x}\right)^2. $
negative_passages.docid: 101708
negative_passages.text: Integers can be expressed as $a^3+b^3+c^3-3abc$
negative_passages.docid: 69673
negative_passages.text: Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$
negative_passages.docid: 115513
negative_passages.text: Find the minimum of $\sqrt{(x+1)^2+(y-1)^2}+\sqrt{(x+2)^2+(y+2)^2}+\sqrt{(x-1)^2+(y+1)^2}$ For real $x,y$
negative_passages.docid: 115850
negative_passages.text: Minimum of $|ax-by+c|$
negative_passages.docid: 66261
negative_passages.text: Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$
negative_passages.docid: 39179
negative_passages.text: Why is $(0, 0)$ not a minimum of $f(x, y) = (y-3x^2)(y-x^2)$?
negative_passages.docid: 99692
negative_passages.text: Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$
negative_passages.docid: 23080
negative_passages.text: Prove that $a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc.$
negative_passages.docid: 131168
negative_passages.text: Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$
negative_passages.docid: 62758
negative_passages.text: How find this value $\frac{a^3}{(a-p)(a-q)}+\frac{b^3}{(b-p)(b-q)}+\frac{c^3}{(c-p)(c-q)}$
negative_passages.docid: 76638
negative_passages.text: Find the limit $ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$
negative_passages.docid: 137864
negative_passages.text: Prove that $ \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.
negative_passages.docid: 140001
negative_passages.text: Minimum value of $2^{\sin^2x}+2^{\cos^2x}$