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

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

#Algebra #SystemOfNonLinearEquations #GCDC #CyclicSystem

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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Comments

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 - 6 years, 6 months ago

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 - 6 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$