query: Algebra Problem: $a + 1/b = b + 1/c = c + 1/a = t $
positive_passages.docid: 20957
positive_passages.text: If $a+\frac1b=b+\frac1c=c+\frac1a$ for distinct $a$, $b$, $c$, how to find the value of $abc$?
negative_passages.docid: 124742
negative_passages.text: P(A|C)=P(A|B)*P(B|C)?
negative_passages.docid: 109745
negative_passages.text: If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
negative_passages.docid: 142076
negative_passages.text: $c$ is a complex number that satisyfing $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$
negative_passages.docid: 5149
negative_passages.text: Show $\binom{a}{b}\binom{b}{c}=\binom{a}{a-c}\binom{a-c}{a-b}$
negative_passages.docid: 112969
negative_passages.text: Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
negative_passages.docid: 113641
negative_passages.text: Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)
negative_passages.docid: 70311
negative_passages.text: Trying to prove (A ∩ B) − C = (A − C) ∩ (B − C)
negative_passages.docid: 76680
negative_passages.text: Prove that if $\gcd(a,c)=1$ and $\gcd(b,c)=1$ then $\gcd(ab,c)=1$
negative_passages.docid: 137626
negative_passages.text: If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
negative_passages.docid: 26861
negative_passages.text: $a+b=c \times d$ and $a\times b = c + d$
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: 72403
negative_passages.text: Given $A+B+C=180^{\circ}$, find value of $\tan A\cdot\tan B+\tan B\cdot\tan C+\tan A\cdot \tan C-\sec A\cdot\sec B\cdot\sec C$
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: 124868
negative_passages.text: Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
negative_passages.docid: 128820
negative_passages.text: Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
negative_passages.docid: 95852
negative_passages.text: Prove that $(a+b)(b+c)(c+a) \ge8$