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Question

Let a and b be two positive real numbers. Suppose A1,A2 are two arithmetic means; G1,G2 are two geometric means and H1,H2 are two harmonic means between a and b, then-

A
G1G2H1H2=A1+A2H1+H2
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B
G1G2H1H259=29(ab+ba)
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C
H1+H2A1+A2=9ab(2a+b)(a+2b)
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D
G1G2H1H2=H1+H2A1+A2
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Solution

The correct options are
A G1G2H1H2=A1+A2H1+H2
B G1G2H1H259=29(ab+ba)
D H1+H2A1+A2=9ab(2a+b)(a+2b)
A1=a+13(ba),A2=b+23(ba)
A1+A2=a+b
Similarly, G1=a(ba)1/3,G2=a(ba)2/3
G1G2=ab
and
1H1=1a+13(1b1a),
1H2=1a+23(1b1a)
Now, 1H1+1H2=1a+1b
H1+H2H1H2=a+bab=A1+A2G1G2
G1G2H1H2=A1+A2H1+H2
Now, H1+H2=3aba+2b+3ab2a+b=9ab(a+b)(a+2b)(2a+b)
A1+A2H1+H2=2(a2+b2)+5ab9ab
Thus, G1G2A1A259=29(ab+ba)
Hence, option A, B and C are correct.

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