query: Find the minimum value of $(a+\frac{1}{b})^2 + (b+\frac{1}{c})^2 + (c+\frac{1}{a})^2$
positive_passages.docid: 26617
positive_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: 129052
negative_passages.text: Find all functions satisfying $\frac{1}{f(a)}+ \frac{1}{f(b)} = \frac{1}{f(c)} $ whenever $\frac{1}{a}+ \frac{1}{b} = \frac{1}{c} $
negative_passages.docid: 104922
negative_passages.text: Inequality $\frac1a + \frac1b + \frac1c \leq \frac{a^8+b^8+c^8}{a^3b^3c^3}$
negative_passages.docid: 155493
negative_passages.text: Minimum value of the expression given below.
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: 134159
negative_passages.text: Smallest value of $a^2 + b^2 + c^2+ d^2$, given values for $(a+b)(c+d)$, $(a+c)(b+d)$, and $(a+d)(b+c)$
negative_passages.docid: 40779
negative_passages.text: What is the smallest possible value of $\lfloor (a+b+c)/d\rfloor+\lfloor (a+b+d)/c\rfloor+\lfloor (a+d+c)/b\rfloor+\lfloor (d+b+c)/a\rfloor$?
negative_passages.docid: 90902
negative_passages.text: Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.
negative_passages.docid: 41792
negative_passages.text: If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $
negative_passages.docid: 118507
negative_passages.text: Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
negative_passages.docid: 26287
negative_passages.text: Find $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ if $a+b+c=0$
negative_passages.docid: 35719
negative_passages.text: Prove that $\frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{c^2+1} \leq \frac{9}{10}$ if $a+b+c=1$.
negative_passages.docid: 92966
negative_passages.text: All integer values of $\frac{a^2+b^2+1}{ab-1}$
negative_passages.docid: 115850
negative_passages.text: Minimum of $|ax-by+c|$
negative_passages.docid: 3885
negative_passages.text: Maximum and minimum of of $f(x)=|x-1|+|x-2|+|x-3|$
negative_passages.docid: 29424
negative_passages.text: Maximum value of $\sin A+\sin B+\sin C$?
negative_passages.docid: 106552
negative_passages.text: Find the sum of $\sum \frac{1}{k^2 - a^2}$ when $0<a<1$