I self-realized an interesting property today that all numbers $(a,b)$ belonging to the infinite set $$\{(a,b): a=(2l+1)^2, b=(2k+1)^2;\ l,k \in N;\ l,k\geq1\}$$ have their AM and GM both integers.
Now I wonder if there exist distinct real numbers $(a,b)$ such that their arithmetic mean, geometric mean and harmonic mean (AM, GM, HM) all three are integers. Also, I wonder if a stronger result for $(a,b)$ both being integers exists.
I tried proving it, but I did not find it easy. For the AM, it is easy to assume a real $a$ and an AM $m_1$ such that the second real $b$ equals $2m_1-a$. For the GM, we get a condition that $m_2=\sqrt{(2m_1-a)a}$. If $m_2$ is an integer, then… what? I am not sure exactly how we can restrict the possible values of $a$ and $m_1$ in this manner.
Best Answer
Expanding on Christian Blatter's answer.
There are a few key points.
These key points lead to a strategy for finding numbers whose am, gm and hm are all integers.
Now to work this through, pick any two distinct positive integers $x$ and $y$.
$$\mathrm{GM}(x^2,y^2) = xy$$ $$\mathrm{AM}(x^2,y^2) = \frac{x^2+y^2}{2}$$ $$\mathrm{HM}(x^2,y^2) = \frac{2x^2y^2}{x^2+y^2}$$
Let $t = 2(x^2 + y^2)$ Let $a=tx^2$ Let $b=ty^2$. Since only addition, multiplication and squaring of positive integers is involved it is clear that $t$, $a$ and $b$ are all positive integers. It is also clear that a and b are distinct.
$$\mathrm{GM}(a,b) = txy$$ $$\mathrm{AM}(a,b) = t\frac{x^2+y^2}{2} = (x^2+y^2)^2$$ $$\mathrm{HM}(a,b) = t\frac{2x^2y^2}{x^2+y^2} = 4x^2y^2$$
Again since all these values can be calculated merely by adding, multiplying and squaring positive integers they are all positive integers.
Lets plug in some numbers, for example $x=1$ and $y=2$
$$t = 10$$ $$a = 10$$ $$b = 40$$ $$\mathrm{GM}(10,40) = 20$$ $$\mathrm{AM}(10,40) = 25$$ $$\mathrm{HM}(10,40) = 16$$
Indeed we can extend this techiquie to find an arbitary size list of integers the AM, GM, and HM of any subset of which are integers. Just start with integers of the form $x^{n!}$ so the GMs are all integers. Then work out the AMs and HMs and multiply through.