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Grotesque

Consider the following system of equations:

\[a = (\tfrac{15b}{c} + 5d - 1)^{0.5} , \; b = (\tfrac{15c}{d} + 5a - 1)^{0.5} \] \[c = (\tfrac{15d}{a} + 5b -1 )^{0.5}, \; d =(\tfrac{15a}{b} + 5c - 1)^{0.5} \]

Evaluate the value(s) of the expression

\[\dfrac{abc+abd+acd+bcd+4abcd}{a+b+c+d+ab+ac+ad+bc+bd+cd.}\]

Note by Guilherme Dela Corte
3 years, 6 months ago

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Hint: if we let \(a=b=c=d\), we yield \(a=7\), and the desired expression has only one possible value: \(\boxed{14.}\)

Guilherme Dela Corte - 3 years, 1 month ago

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You can't just assume that the expression is constant \(\forall a,b,c,d\in\mathbb R\); you'll have to prove it.

Mathh Mathh - 3 years, 1 month ago

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