query: $|a|+|b|+|c|-|a+b|-|b+c|-|a+c|+|a+b+c| \geq 0 ? \forall a,b,c,d \in \Bbb {C} $
positive_passages.docid: 80095
positive_passages.text: Show $|a|+|b|+|c|+|a+b+c| \geq  |a+b|+|b+c|+|c+a|$ for complex $a$, $b$, $c$
negative_passages.docid: 115504
negative_passages.text: Find the value of a+b+c
negative_passages.docid: 60122
negative_passages.text: Prove $(1+a+b)(1+b+c)(1+c+a)\ge 9(ab+bc+ca)$
negative_passages.docid: 99692
negative_passages.text: Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$
negative_passages.docid: 68986
negative_passages.text: Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$
negative_passages.docid: 83996
negative_passages.text: For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
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: 23264
negative_passages.text: Inequality $ \vert \sqrt{a}-\sqrt{b} \vert \leq \sqrt{ \vert a -b \vert } $
negative_passages.docid: 127296
negative_passages.text: prove that a|b and b|a iff a=+-b
negative_passages.docid: 115719
negative_passages.text: Find the possible values of |A + B + C |
negative_passages.docid: 70311
negative_passages.text: Trying to prove (A ∩ B) − C = (A − C) ∩ (B − C)
negative_passages.docid: 28499
negative_passages.text: Inequality.$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq 3$
negative_passages.docid: 137864
negative_passages.text: Prove that $ \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.
negative_passages.docid: 18119
negative_passages.text: Show that if a,b,c,d are positive then $\frac{ab}{a+b}+\frac{cd}{c+d}\le \frac{(a+c)(b+d)}{a+b+c+d}$
negative_passages.docid: 35242
negative_passages.text: If $a$, $b$, $c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$, and $\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are all integers, then $|a|=|b|=|c|$
negative_passages.docid: 97667
negative_passages.text: If $a<b$, $c<d$ then $a+c<b+d$?
negative_passages.docid: 29935
negative_passages.text: If $a, b, c, d>0$ and $abcd=1$ prove that an inequality holds true