query: Lower bound for cyclic expression in four variables
positive_passages.docid: 98454
positive_passages.text: Prove that, $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\ge 4$ where we do not use AM-GM inequality on the given statement to prove it.
negative_passages.docid: 152311
negative_passages.text: Proving expression$\geq 2$, where variables are $x,y,z$
negative_passages.docid: 62501
negative_passages.text: Minimum value of $\frac{4}{4-x^2}+\frac{9}{9-y^2}$
negative_passages.docid: 114146
negative_passages.text: Polynomial in $n$ variables
negative_passages.docid: 90537
negative_passages.text: Converse of Fermat's Little Theorem.
negative_passages.docid: 138304
negative_passages.text: getting tight bounds on n!
negative_passages.docid: 33117
negative_passages.text: Cyclic containment of sets
negative_passages.docid: 124901
negative_passages.text: Big-O notation and polynomials
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: 39504
negative_passages.text: Why does $ \frac{2x}{2+x}$ provide a particularly tight lower bound for $\ln(1+x)$ for small positive values of $x$?
negative_passages.docid: 152506
negative_passages.text: The greatest common divisor is the smallest positive linear combination
negative_passages.docid: 74315
negative_passages.text: Constraint on variables in summation
negative_passages.docid: 25744
negative_passages.text: Multiplication in Permutation Groups Written in Cyclic Notation
negative_passages.docid: 138099
negative_passages.text: A cyclic inequality $\sum\limits_{cyc}{\sqrt{3x+\frac{1}{y}}}\geqslant 6$
negative_passages.docid: 94327
negative_passages.text: Cubic Diophantine with two variables
negative_passages.docid: 20659
negative_passages.text: Is trace invariant under cyclic permutation with rectangular matrices?
negative_passages.docid: 118674
negative_passages.text: Simple upper bound for $\binom{n}{k}$