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Question

Let f(x)=0 be a cubic equation with positive and distinct roots α,β,γ such that β is harmonic mean between the roots of f(x)=0. If r=[2βα+γ]+[2αγαβ+βγ], then the value of 3i=1ir is
( Here, [.] denotes the greatest integer function.)

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Solution

Let f(x)=x3+ax2+bx+c=0
and α,β,γ be the roots of the above equation.
α+β+γ=a
αβ+βγ+γα=b
αβγ=c

f(x)=3x2+2ax+b
Let x1,x2 be the roots of f(x)=0
x1+x2=2a3
x1x2=b3
Given, β is harmonic mean between x1 and x2.
β=2x1x2x1+x2=2b2a=ba
β=(αβ+βγ+γα)(α+β+γ)
αβ+β2+βγ=αβ+βγ+γα
β2=αγ
α,β,γ are in G.P.

r=[2βα+γ]+[2αγαβ+βγ]
r=[2αγα+γ]+[2αγβ(α+γ)]
r=[G.M.A.M.]+[H.M.G.M.]=0+0=0
{A.M.>G.M.>H.M.}
ri=1(i)3=3i=11=1+1+1=3

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