#System for a,b,c positive

17 messages · Page 1 of 1 (latest)

slim nexus
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Hey , can you help for this system :
a+b+c=3
(a/b)^2+(b/c)^2+(c/a)^2=3

I was thinking of substitution but I can't find the good one...

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They are not only positive my bad

slim nexus
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I found the 4 (a,b,c) solutions 😄

north radish
slim nexus
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I used inequalities of average

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Let x = a/b ; y = b/c and z = c/a

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now we have x^2+y^2+z^2=3

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So the quadratic average is equal to 1

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and we can notice that xyz=1

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so the geometric average is also 1

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and with average inequalities we know that GM<AM<QM

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so (x+y+z)/3 = 1

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The only solutions for (a,b,c) are (1,1,1) (3,3,-3) (3,-3,3) (-3,3,3)

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the other method is using number theory :

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we want the sum of 3 fractions be equal to 3

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so a^2=b^2=c^2

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This is enough to conclude 😄