query: Prove that $a^2b^2+b^2c^2+a^2c^2\leq 3$.
positive_passages.docid: 23638
positive_passages.text: $a^2b^2+a^2c^2+b^2c^2\leq 3$
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: 152053
negative_passages.text: Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
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: 130281
negative_passages.text: Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
negative_passages.docid: 126018
negative_passages.text: Show that $ e^{A+B}=e^A e^B$
negative_passages.docid: 52625
negative_passages.text: Prove inequality $\sqrt{\frac{2a}{b+a}} + \sqrt{\frac{2b}{c+b}} + \sqrt{\frac{2c}{a+c}} \leq 3$
negative_passages.docid: 74222
negative_passages.text: If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$
negative_passages.docid: 48484
negative_passages.text: Prove that $6|a+b+c$ if and only if $6|a^3+b^3+c^3$
negative_passages.docid: 60073
negative_passages.text: Proving $4(a^3 + b^3) \ge (a + b)^3$ and $9(a^3 + b^3 + c^3) \ge (a + b + c)^3$
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: 52204
negative_passages.text: Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$
negative_passages.docid: 89184
negative_passages.text: Prove $30|(a^3b-ab^3) $
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: 130327
negative_passages.text: Prove that $a^2 > b^2$ ; if $a > b \geq 0$.
negative_passages.docid: 90916
negative_passages.text: Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.
negative_passages.docid: 65280
negative_passages.text: If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $.