query: Inequality: $x^2+y^2+xy\ge 0$
positive_passages.docid: 27911
positive_passages.text: How can I prove that $xy\leq x^2+y^2$?
negative_passages.docid: 56391
negative_passages.text: Is this inequality true? $ (x + y)^{\alpha} < x^{\alpha} + y^{\alpha} $, for positive $x$ & $y$, and for $0 < \alpha < 1$
negative_passages.docid: 108704
negative_passages.text: Help with proof of $|x+y| \geq |x|-|y|$
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: 137522
negative_passages.text: Hard inequality $ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $
negative_passages.docid: 8207
negative_passages.text: Can I prove this inequality algebraically?
negative_passages.docid: 56101
negative_passages.text: Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
negative_passages.docid: 32149
negative_passages.text: Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$
negative_passages.docid: 158276
negative_passages.text: Inequality problem involving square roots
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 139711
negative_passages.text: prove this inequality with $\sum\limits_\text{cyc}\sqrt{1-xy}\ge 2$
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: 14677
negative_passages.text: Proving the exponential inequality: $x^y+y^x\gt1$
negative_passages.docid: 66261
negative_passages.text: Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$
negative_passages.docid: 66996
negative_passages.text: Proof by induction of Bernoulli's inequality $ (1+x)^n \ge 1+nx$
negative_passages.docid: 47644
negative_passages.text: Inequality involving rearrangement: $ \sum_{i=1}^n |x_i - y_{\sigma(i)}| \ge \sum_{i=1}^n |x_i - y_i|. $
negative_passages.docid: 5765
negative_passages.text: Show that $[2x]+[2y] \geq [x]+[y]+[x+y]$