query: If $a^{3}+b^{3}+c^{3}=(a+b+c)^{3},$ prove that $a^{5}+b^{5}+c^{5}=(a+b+c)^{5}.$ Generalize your result.
positive_passages.docid: 149759
positive_passages.text: Problem on multiplication formulae.
negative_passages.docid: 5729
negative_passages.text: If $a^2+b^2+c^2=1$ then prove the following.
negative_passages.docid: 118507
negative_passages.text: Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
negative_passages.docid: 11868
negative_passages.text: Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$
negative_passages.docid: 150320
negative_passages.text: Why does $a^{b}·a^{c}=a^{b+c}$?
negative_passages.docid: 146753
negative_passages.text: Find $x$ such that $(x^2 + 4x + 3)^x + (2x + 4)^x = (x^2 + 4x + 5)^x$
negative_passages.docid: 113789
negative_passages.text: Given that$(3x-1)^7=a_7x^7+a_6x^6+a_5x^5+...+a_1x+a_0$, find $a_7+a_6+a_5+a_4+...+a_1+a_0$
negative_passages.docid: 87304
negative_passages.text: Solving for $x$ : $a^x+b^x=c$
negative_passages.docid: 111295
negative_passages.text: Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$
negative_passages.docid: 4017
negative_passages.text: factorize $x^5+ax^3+bx^2+cx+d$ if $d^2+cb^2=abd$
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
negative_passages.docid: 115460
negative_passages.text: Proving $a\mid b^2,\,b^2\mid a^3,\,a^3\mid b^4,\ldots\implies a=b$ - why is my approach incorrect
negative_passages.docid: 81136
negative_passages.text: Prove by induction that... $1+3+5+7+...+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$
negative_passages.docid: 142263
negative_passages.text: Assuming $ab^2 = b^3a$ and $a^2=1$ prove that the order of $b$ is $5$.
negative_passages.docid: 2289
negative_passages.text: Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
negative_passages.docid: 44247
negative_passages.text: For $a^3+b^3+c^3=3$ prove that $a^4b+b^4c+c^4a\leq3$
negative_passages.docid: 96327
negative_passages.text: Show that $a^x+b^x+c^x>(a+b+c)^x$ for all $a,b,c>0$ and $0<x<1$