query: Find $a^3+b^3+c^3-3abc$ (binomial theorem)
positive_passages.docid: 126385
positive_passages.text: How to prove $a^3 + b^3 +c^3 - 3abc =1 $, If a,b,c are infinite series as given in the description box below?
negative_passages.docid: 19940
negative_passages.text: How to show that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ indirectly?
negative_passages.docid: 29963
negative_passages.text: $6$ divides $a+b+c$ if and only if $6$ divides $a^{3}+b^{3}+c^{3}$
negative_passages.docid: 39199
negative_passages.text: closed form for $\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}$
negative_passages.docid: 139082
negative_passages.text: If $a,b,c>0$ and $a^3+b^3=c^3$ then prove that $a^2+b^2-c^2>6(c-a)(c-b)$.
negative_passages.docid: 44621
negative_passages.text: Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$
negative_passages.docid: 87304
negative_passages.text: Solving for $x$ : $a^x+b^x=c$
negative_passages.docid: 62758
negative_passages.text: How find this value $\frac{a^3}{(a-p)(a-q)}+\frac{b^3}{(b-p)(b-q)}+\frac{c^3}{(c-p)(c-q)}$
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: 156468
negative_passages.text: Solutions to $a+b+c=12$, $a, b, c \in \mathbb{N}_0$
negative_passages.docid: 23638
negative_passages.text: $a^2b^2+a^2c^2+b^2c^2\leq 3$
negative_passages.docid: 116491
negative_passages.text: Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$
negative_passages.docid: 95643
negative_passages.text: Proving $\sum_{cyc} \sqrt{a^2+ab+b^2}\geq\sqrt 3$ when $a+b+c=3$
negative_passages.docid: 19402
negative_passages.text: Find integral solution of $a^{b} - b^{a} = 3$
negative_passages.docid: 41792
negative_passages.text: If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $
negative_passages.docid: 89184
negative_passages.text: Prove $30|(a^3b-ab^3) $
negative_passages.docid: 26287
negative_passages.text: Find $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ if $a+b+c=0$