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# Let be positive real numbers such that Find the minimum value of

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Let   $$a,b,c,d$$  be positive real numbers such that $$a+b+c+d=1$$ Find the minimum value of

$$\frac{a}{b + c + d} + \frac{b}{a + c + d} + \frac{c}{a + b + d} + \frac{d}{a + b + c}.$$

Aug 3, 2019

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using Lagrange multipliers you can show that the minimum occurs when a=b=c=d=1/4 and is equal to 4/3

but you can also argue this from symmetry since they are all required to be positive.

Aug 3, 2019