query: For positive real numbers $a,b,c$ prove that $ a^4 + b^4 + c^4 \ge abc(a+b+c)$
positive_passages.docid: 148433
positive_passages.text: Using arithmetic mean>geometric mean
negative_passages.docid: 44705
negative_passages.text: Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$
negative_passages.docid: 5465
negative_passages.text: This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$
negative_passages.docid: 116933
negative_passages.text: Let $a,b,c$ be non-negative real numbers , then $\frac{(a+b+c)^2}{3}\ge a\sqrt{bc}+b\sqrt{ca}+c\sqrt{ab}$?
negative_passages.docid: 25870
negative_passages.text: Prove that $0 \leq ab + ac + bc - abc \leq 2.$
negative_passages.docid: 4333
negative_passages.text: Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$
negative_passages.docid: 25258
negative_passages.text: Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $.
negative_passages.docid: 26741
negative_passages.text: How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
negative_passages.docid: 98305
negative_passages.text: How do I show that $\sum_{cyc} \frac {a^6}{b^2 + c^2} \ge \frac {abc(a + b + c)}2?$
negative_passages.docid: 157416
negative_passages.text: Prove inequality: $\sum \frac{a^4}{a^3+b^3} \ge \frac{a+b+c}{2}$
negative_passages.docid: 134535
negative_passages.text: Inequality $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c$ when $abc = 1$
negative_passages.docid: 34275
negative_passages.text: How do I show that $\frac {a^2}b + \frac {b^2}c + \frac {c^2}d + \frac {d^2}a \ge 4$ for $a^2 + b^2 + c^2 + d^2 = 4$?
negative_passages.docid: 5694
negative_passages.text: How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$?
negative_passages.docid: 41558
negative_passages.text: Prove that $\frac4{abcd} \geq \frac a b + \frac bc + \frac cd +\frac d a$
negative_passages.docid: 12954
negative_passages.text: Prove if $ac>bc$ and $c>0$, then $a>b$
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: 118467
negative_passages.text: Prove that $a^2+ab+b^2\ge 0$