query: If $a,b,c$ are positive real numbers and $(1+a+b+c)\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=16$, then $a+b+c=3$
positive_passages.docid: 16140
positive_passages.text: Condition on a,b and c satisfying an equation(TIFR GS 2017)
negative_passages.docid: 55975
negative_passages.text: Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$.
negative_passages.docid: 21726
negative_passages.text: Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
negative_passages.docid: 20878
negative_passages.text: If $(a,b)=1$ then prove $(a+b, ab)=1$.
negative_passages.docid: 72882
negative_passages.text: If $a$, $b$, and $c$ are sides of a triangle, then $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$.
negative_passages.docid: 59239
negative_passages.text: For non-zero real $x$, $y$, $z$ and complex $a$, $b$, $c$ with $|a|=|b|=|c|$, if $x+y+z=0=ax+by+cz$, then $a=b=c$
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: 33998
negative_passages.text: Show that given seven real numbers, it is always possible take two of them, such that $\left\vert\frac{a-b}{1+ab}\right\vert<\frac{1}{\sqrt{3}}$
negative_passages.docid: 11127
negative_passages.text: Let $a$, $b$, and $c$ be positive real numbers.
negative_passages.docid: 17264
negative_passages.text: Given $a,b,c$ are the sides of a triangle. Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
negative_passages.docid: 141525
negative_passages.text: Cyclic Inequality $\frac{a^3}{(a+b)(a+c)}$
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: 48484
negative_passages.text: Prove that $6|a+b+c$ if and only if $6|a^3+b^3+c^3$
negative_passages.docid: 156541
negative_passages.text: $a^b > b^a$ or $a^b b$ where $a,b$ are positive real numbers
negative_passages.docid: 52305
negative_passages.text: Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b,c > 0 $?
negative_passages.docid: 94831
negative_passages.text: Finding value of $a +b+c$
negative_passages.docid: 33240
negative_passages.text: $ \frac{1}{a^{2} + b^{2} + 1} + \frac{1}{b^{2}+c^{2}+1} + \frac{1}{c^{2} + a^{2} + 1} \ge 1 $, prove $ab + bc + ca \le 3$