query: Algebraic inequalities with equality condition
positive_passages.docid: 95852
positive_passages.text: Prove that $(a+b)(b+c)(c+a) \ge8$
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: 12837
negative_passages.text: Inequality with condition $x^2+y^2+z^2=1$.
negative_passages.docid: 25203
negative_passages.text: with inequality $\frac{y}{xy+2y+1}+\frac{z}{yz+2z+1}+\frac{x}{zx+2x+1}\le\frac{3}{4}$
negative_passages.docid: 64955
negative_passages.text: Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
negative_passages.docid: 85039
negative_passages.text: Purely "algebraic" proof of Young's Inequality
negative_passages.docid: 45918
negative_passages.text: Inequality: $x^2+y^2+xy\ge 0$
negative_passages.docid: 136290
negative_passages.text: a mathematical problem on inequalities
negative_passages.docid: 134049
negative_passages.text: Showing an inequality is valid
negative_passages.docid: 142360
negative_passages.text: How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
negative_passages.docid: 54703
negative_passages.text: Find the a < 0 for which the inequalities have at least one common solutions
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$
negative_passages.docid: 33332
negative_passages.text: A classic exponential inequality: $x^y+y^x>1$
negative_passages.docid: 30386
negative_passages.text: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
negative_passages.docid: 151148
negative_passages.text: How to solve simultaneous inequalities?
negative_passages.docid: 125028
negative_passages.text: Prove the following equality using mathematical induction:
negative_passages.docid: 132595
negative_passages.text: $\sup_{0 \leq M \leq I_{\mathcal{H}_1}} \frac{\operatorname{Tr}[\rho M]}{\operatorname{Tr}[\sigma M]}$ satisfies Data Processing Inequality