Find $m + n$ if $m$ and $n$ are natural numbers such that:$$\frac {m+n} {m^2+mn+n^2} = \frac {4} {49}\;.$$
Say: $$\begin{cases}m+n = 4k\\ m^2+mn+n^2 = 49k\end{cases}$$
It follows: \begin{align*} (m+n)^2 = (4k)^2 & = 16k^2\\ m^2+mn+n^2 + mn & = 16k^2\\ mn & = 16k^2 - 49k \end{align*}
So \begin{align*} mn & \gt 0\\ 16k^2 - 49k & \gt 0\\ k & \gt 3 \end{align*}
Then no more progress.
A simple approach is to use the symmetry and choose $m \ge n$, then note that $m$ cannot be very large. If it is $13$ or more the fraction will be too small: $\frac {m+n}{m^2+mn+n^2} \lt \frac {m+n}{m^2+mn} = \frac 1m$. That doesn't leave many choices. Now the quickest is to make a spreadsheet with $m$ across the top, $n$ down the side, compute $\frac {m+n}{m^2+mn+n^2} -\frac 4{49}$, and scan by eye for zeros. Alternately, you can make it a quadratic $4m^2+4mn+4n^2-49m-49n=0$ and insert the $n$'s and solve for $m$. See when you get naturals.
Observe that $k$ must be a non-zero integer.
We know that $m, n$ are the roots of the quadratic equation
$$X^2 - 4kX + (16k^2 - 49k)$$
The roots, from the quadratic equation, are
$$ \frac { 4k \pm \sqrt{(4k)^2 - 4(16k^2 - 49k) }} {2} = 2k \pm \sqrt{ 49k - 12k^2}$$
The expression in the square root must be a perfect square.
Try $k = 1$, $49 k - 12k^2 = 37$ is not a perfect square.
Try $k = 2$, $49k - 12k^2 = 50$ is not a perfect square.
Try $k=3$, $49k-12k^2 = 39$ is not a perfect square.
Try $k=4$, $49k-12k^2 = 4$ is a perfect square. This leads to roots 6, 10, which have sum 16.
For $k\geq 5$, $49k - 12k^2 < 0$ has no solution.
For $k \leq -1$, $49k - 12k^2 < 0$ has no solution.
$4m^2+4mn+4n^2=49m+49n$
$4m^2+4mn+n^2+3n^2-49m-49n=0$
$(2m+n)^2+3n^2-(49/2)(2m+n)-(49/2)n=0$
$16(2m+n)^2-392(2m+n)+48n^2-392n=0$
$(4(2m+n)-49)^2+48n^2-392n=2401$
$3(4(2m+n)-49)^2+144n^2-1176n=7203$
$3(4(2m+n)-49)^2+(12n-49)^2=9604$
$3(8m+4n-49)^2+(12n-49)^2=9604$.
$3x^2+y^2=98^2$
Now the last equation has only finitely many integer solutions, with a finite procedure for finding them, then for each solution you can check whether $8m+4n-49=x,12n-49=y$ has integer solutions, and that should do it.
Finishing the original reasoning started in the question:
$$(\frac {m+n}{2})^2 -mn \ge0 \Rightarrow (\frac {4k}{2})^2 - (16k^2 -49k) \ge0 $$ $$ 49k - 12k^2 \ge 0 \Rightarrow k\le4$$
If: $$3\lt k \le 4 \Rightarrow k=4$$
Finally, substituting $k=4$ in $m+n = 4k$, we get $m+n=16$.
It is useful to write the stipulation of the problem in a few different ways: $$\frac{49}{4}=\frac{(m+n)^2-mn}{m+n}=m+n-\frac{mn}{m+n},$$ $$\frac{49}{4}=\frac{(m+n)n+m^2}{m+n}=n+\frac{m^2}{m+n},$$ $$\frac{49}{4}=\frac{(m+n)m+n^2}{m+n}=m+\frac{n^2}{m+n}.$$ From the first, we glean that $m+n>12$ because $49/4=12+1/4$ and $m,n>0$. From the second and third, we see that $n\le 12$ and $m\le 12$.
Another thing to notice is that $mn/(m+n)$ must reduce to a fraction with a denominator of $4$. This means that $$4mn=c(m+n)$$ for some $c$ relatively prime to $4$. This implies that $4$ divides $m+n$, so there are only three cases to consider: $m+n=16$, $m+n=20$, $m+n=24$. From the first, we have $$12+\frac{1}{4}=16-\frac{m(16-m)}{16}$$ which easily gives the solution $\{m,n\}=\{6,10\}$. The other two cases result in quadratic equations with no real solutions.
