query: Prove that $0 \le xy+yz+xz - 2xyz \le 7/27, \:\:\:x,y,z \ge 0$ with $x+y+z=1$.
positive_passages.docid: 157193
positive_passages.text: IMO 1984: Prove that $0 ≤ yz + zx +xy −2xyz ≤ \frac {7}{27}$, where $x,y$ and $z$ are non-negative real numbers for which $x + y + z = 1.$
negative_passages.docid: 50433
negative_passages.text: Prove $4(x + y + z)^3 > 27(x^2y + y^2z + z^2x)$
negative_passages.docid: 118467
negative_passages.text: Prove that $a^2+ab+b^2\ge 0$
negative_passages.docid: 129331
negative_passages.text: Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
negative_passages.docid: 150565
negative_passages.text: Prove $(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}$
negative_passages.docid: 125817
negative_passages.text: show this inequality $(x+y)^3+(y+z)^3+(z+w)^3+(w+x)^3\ge 8(x^2y+y^2z+z^2w+w^2x)$
negative_passages.docid: 66707
negative_passages.text: How to prove this inequality? $ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $
negative_passages.docid: 95852
negative_passages.text: Prove that $(a+b)(b+c)(c+a) \ge8$
negative_passages.docid: 85917
negative_passages.text: Solve $(x+1)^y-x^z=1$
negative_passages.docid: 54370
negative_passages.text: If $x+y+z=xyz$, prove $\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{2x}{1-x^2}\times\frac{2y}{1-y^2}\times\frac{2z}{1-z^2}$
negative_passages.docid: 139711
negative_passages.text: prove this inequality with $\sum\limits_\text{cyc}\sqrt{1-xy}\ge 2$
negative_passages.docid: 55210
negative_passages.text: Prove that the equation: $c_0+c_1x+\ldots+c_nx^n=0$ has a real solution between 0 and 1.
negative_passages.docid: 120736
negative_passages.text: If xy + yz + zx = 1, ...........
negative_passages.docid: 94119
negative_passages.text: Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
negative_passages.docid: 99676
negative_passages.text: Prove $\sum{\frac{1}{(x+2y)^2}} \geq\frac{1}{xy+yz+zx}$
negative_passages.docid: 92692
negative_passages.text: Prove $3(x^2y+y^2z+z^2x)(xy^2+yz^2+zx^2)\ge xyz(x+y+z)^3$
negative_passages.docid: 41383
negative_passages.text: Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$