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After following Can two sets have same AM, GM, HM? and the sublime answer of Micah, I am tempted ask the solution of the same question when size of these two sets are not same.

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What happens if the "two sets of different size" consist of just one repeated value? – J. M. Jul 19 '12 at 7:48
What are the implications if repetition is allowed and if not? Off course, all the elements of either set should not be equal. – lab bhattacharjee Jul 19 '12 at 7:51
up vote 2 down vote accepted

Sure, why not? The general statement that my answer was a special case of is:

If $\{r_1,\dots,r_n\}$ are the roots of the polynomial $x^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$, then their arithmetic mean is $-a_{n-1}/n$, their geometric mean is $\sqrt[n]{(-1)^na_0}$, and their harmonic mean is $-na_0/a_1$.

This can be shown by examining the Vieta formulae in exactly the same way as the degree-$4$ case, and gives a recipe for building sets of any size at least 3 that have whatever AM, GM, and HM you want. The only possible worry is that the roots might not all be real and positive, in which case strictly speaking the GM isn't well-defined; for example, this will happen if you try to build a set that violates the AM-GM-HM inequalities.

For a concrete-ish example, the set $\{1,2\}$ has AM $3/2$, HM $4/3$, and GM $\sqrt{2}$. By the above formulae, so do the roots of $x^3-9/2x^2+9\sqrt{8}/4-\sqrt{8}$ (which WolframAlpha says are $\left\{\sqrt{2},\frac{1}{4}\left(9-2\sqrt{2}\pm\sqrt{57-36\sqrt{2}}\right)\right\}$).

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