I am currently working on an Olympiad math problem, and I am struggling to find a solution. I would greatly appreciate your help in solving this problem. I was unable to solve the problem because I don't know the exact rules for solving the problem.
Problem Source: Bangladesh Math Olympiad
A small hint will be enough for me to proceed.
In a triangle, $\triangle ABC$, $\angle B=90^\circ$, and every side has a positive integer-valued length. $△DEF$ is inside of $\triangle ABC$ in such a way that $AB\parallel DE$, $BC\parallel EF$, $AC\parallel DF$, and the distance between the parallel sides of the triangles are always $2$. How many triangles can be formed so that the area of $\triangle ABC$ is $9$ times $\triangle DEF$?
I revised this answer to make it more of a hint. I'm sorry I did not originally see your "small help" request.
Let the lengths of $\triangle ABC$ be $a$, $b$, and $c$, with $b$ the hypotenuse. For orientation, let $c$ be horizontal and $a$ be vertical, with $A$ and $C$ the opposite angle.
With $\triangle DEF$, there are corresponding sides. How long is $f$? It is $c$, reduced by $2$ on one side, and using similar triangles, reduced by $\frac{2b}{a}+\frac{2c}{a}$ on the other. (In the picture below, the similar triangle ratios to determine that are left as as exercise.)
So we have one expression for $f$ using subtraction. Meanwhile the two triangles are similar, and with an area ratio of $9$, the legs of the smaller triangle must be $\frac{1}{3}$ as long as of the larger triangle. So you can set up an equation in $a,b,c$ from that.
Next, the big triangle is a Pythagorean triple with $b$ as the hypotenuse, so you can reference the parameterization of such things:
$$\begin{align} b&=k(m^2+n^2)\\ a&=k(m^2-n^2)\\ c&=2kmn \end{align}$$
If you combine this with the equation mentioned above, it leads to a conclusion that there are ________ possibilities and you can list them all.