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Question

Suppose p, q, r are real numbers such that q=p(4p),r=q(4q),p=r(4r). The maximum possible value of p+q+r is?

A
0
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B
3
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C
9
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D
27
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Solution

The correct option is C 9
q=p(4p) ...(i)
r=q(4q) ...(ii)
p=r(4r) ...(iii)

multiplying by pr in (i) by pq in (ii) and by qr in (iii).we get
pqr=p2r(4p) ...(iv)
pqr=q2p(4q) ...(v)
pqr=r2q(4r) ...(vi)

multiplying eq. (iv),(v)and(vi)
(pqr)3=(pqr)3(4p)(4q)(4r)
1=(4p)(4q)(4r) ...(vii)

we know that Arithmatic Mean is always greater or equal to Geometric Mean.
(4p)+(4q)+(4r)33(4p)(4q)(4r)

from eq(vii)

(4p)+(4q)+(4r)331

(4p)+(4q)+(4r)31

(12(p+q+r)31

12(p+q+r)+3

123(p+q+r)

9(p+q+r)

i.e. p+q+r9

so maximum value of p+q+r is 9.

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