query: When $abc=1$ find the minimum of $a^2+b^2+c^2$.
positive_passages.docid: 19909
positive_passages.text: if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$
negative_passages.docid: 73716
negative_passages.text: Let $x,y,z>0$ and $x+y+z=1$, then find the least value of ${{x}\over {2-x}}+{{y}\over {2-y}}+{{z}\over {2-z}}$
negative_passages.docid: 61302
negative_passages.text: Find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$
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: 105164
negative_passages.text: If $abc=1$, prove that $\frac{2}{(a+1)^2+b^2+1} + \frac{2}{(b+1)^2+c^2+1} + \frac{2}{(c+1)^2+a^2+1} \le 1$.
negative_passages.docid: 144017
negative_passages.text: How to find the smallest $n$ such that $n^a\equiv 1 \pmod p$
negative_passages.docid: 78188
negative_passages.text: How can I get the smallest possible answer with the Chinese Remainder Theorem?
negative_passages.docid: 80982
negative_passages.text: Solve $a^3+b^3+3ab=1$ with $(a,b)\in \Bbb{Z}^2$
negative_passages.docid: 82317
negative_passages.text: Find the minimum value of $A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}$
negative_passages.docid: 117821
negative_passages.text: Compute the minimum of a pair of vectors
negative_passages.docid: 31138
negative_passages.text: How to find a global minimum for $\frac{1}{2}x^tQx-b^tx$
negative_passages.docid: 31257
negative_passages.text: Let $a,b,c>0$ such that $a+b+c=3$, find max/min of $\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}$
negative_passages.docid: 216
negative_passages.text: In a $\triangle ABC,a^2+b^2+c^2=ac+ab\sqrt3$,then the triangle is
negative_passages.docid: 158704
negative_passages.text: How to minimize $\sum_{i=1}^p (y_i-x_i)^2$ with constraints $\sum_{i=1}^p y_i - 1 =0$ and $\forall i=\overline{1,p}:-y_i \le 0$?
negative_passages.docid: 134535
negative_passages.text: Inequality $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c$ when $abc = 1$
negative_passages.docid: 152074
negative_passages.text: How to find out the global minimum of the following expression
negative_passages.docid: 155493
negative_passages.text: Minimum value of the expression given below.