query: If $|ax^2+bx+c|\le1$ for $|x|\le1$(redundant?), then prove that $|cx^2+bx+a|\le2$ for $|x|\le1$.
positive_passages.docid: 154945
positive_passages.text: $|px^2+qx+r|\le1$ for all $x$ in $[-1; 1]$, show that $|rx^2+qx+p|\le2$ for all $x$ in $[-1; 1]$
negative_passages.docid: 111847
negative_passages.text: For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
negative_passages.docid: 18606
negative_passages.text: How prove $\frac{a^2}{a+2b^2}+\frac{b^2}{b+2c^2}+\frac{c^2}{c+2a^2}\ge 1$
negative_passages.docid: 5369
negative_passages.text: Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
negative_passages.docid: 64108
negative_passages.text: how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?
negative_passages.docid: 97057
negative_passages.text: Show $a^2+b^2+c^2=1$ $\implies$ $ \dfrac {b^2c^2}{1+a^2} +\dfrac {c^2a^2}{1+b^2}+\dfrac {a^2b^2}{1+c^2} \le \dfrac 14 $
negative_passages.docid: 154327
negative_passages.text: $A^2X = X$ show that $AX = X$, if $A_{ij} > 0$ and $x_{i} > 0$
negative_passages.docid: 56818
negative_passages.text: Prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, x \rangle\geq \alpha|x|^2$ for $x\in\Bbb{R}^n$
negative_passages.docid: 80789
negative_passages.text: Prove that $\forall a,b,c,d\in \mathbb{R}\,$, $(a^2-b^2)(c^2-d^2)\leq(ac-bd)^2$ and equalities hold if and only if $ad=bc$.
negative_passages.docid: 21788
negative_passages.text: Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
negative_passages.docid: 91877
negative_passages.text: Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
negative_passages.docid: 16774
negative_passages.text: If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.
negative_passages.docid: 141102
negative_passages.text: If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.
negative_passages.docid: 25825
negative_passages.text: Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
negative_passages.docid: 133228
negative_passages.text: Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
negative_passages.docid: 33240
negative_passages.text: $ \frac{1}{a^{2} + b^{2} + 1} + \frac{1}{b^{2}+c^{2}+1} + \frac{1}{c^{2} + a^{2} + 1} \ge 1 $, prove $ab + bc + ca \le 3$
negative_passages.docid: 81618
negative_passages.text: Prove $||a|-|b| | \leq |a-b|$