query: Prove that, for all positive integers $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$
positive_passages.docid: 1305
positive_passages.text: Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$
negative_passages.docid: 31295
negative_passages.text: $x+y\sqrt{2}$ infimum ($x,y\in \mathbb{Z}$)
negative_passages.docid: 41293
negative_passages.text: Prove the inequalities $|z_1 +z_2| \geq \frac{1}{2}(|z_1|+|z_2|)|\frac {z_1} {|z_1|} +\frac {z_2} {|z_2|}|$
negative_passages.docid: 129847
negative_passages.text: Show that $\forall (x,y)$ in the first quadrant: $\frac {x^2+y^2}{4}\leq e^{x+y-2}$
negative_passages.docid: 7105
negative_passages.text: Prove that $\prod\limits_{i=1}^n \frac{2i-1}{2i} \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \Bbb Z_+$
negative_passages.docid: 118559
negative_passages.text: Prove the inequality $\sum_{1\le i<j\le n}\sqrt{a_ia_j}<\frac{n-1}2\sum_{i=1}^n a_i$
negative_passages.docid: 126165
negative_passages.text: Prove $|\arctan x - \arctan y|\leq |x-y|$
negative_passages.docid: 122944
negative_passages.text: Prove $|y''(x)| \leq 40$ for all $x \in [1,3]$.
negative_passages.docid: 28218
negative_passages.text: Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
negative_passages.docid: 139711
negative_passages.text: prove this inequality with $\sum\limits_\text{cyc}\sqrt{1-xy}\ge 2$
negative_passages.docid: 72470
negative_passages.text: Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
negative_passages.docid: 92300
negative_passages.text: For positive $a$, $b$, $c$ with $abc=1$, show that $\sum_{cyc}\sqrt{a^2-a+1}\geq a+b+c$
negative_passages.docid: 41364
negative_passages.text: How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?
negative_passages.docid: 65297
negative_passages.text: Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$.
negative_passages.docid: 15116
negative_passages.text: Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$
negative_passages.docid: 80920
negative_passages.text: Prove or disprove: For any $v\in\mathbb{R}^n, \|v\|_1\|v\|_\infty \leq \frac{1+\sqrt{n}}{2}\|v\|_2^2$
negative_passages.docid: 32963
negative_passages.text: proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$