Let $a_1a_2a_3...a_n$ be an n-gon, such that the length of every side is a natural number. Find the length of each side, if it is known that 8 times the side is equal to the square of the sum of the side on its left and the side on it's right sides that are its neighbors. meaning that if a, b, c are consecutive sides, and b is in the middle, $8b=(a+c)^2$.
So I've started with a triangle, and got the solution 2,2,2 but I don't know how to eventually come to n sides. Help?
Starting with a triangle was a good idea. You should be able to see that having every side $2$ will work for any number of sides. Now if you trust the problem setter to give you a problem with a unique answer you are done. Otherwise we need to prove that there is no other solution.
From $(a+c)^2=8b$ we know that $a+c$ must be a multiple of $4$ and $b$ must be even, so all sides are even. Let $a$ be one of the sides of maximal length and $c$ be shorter than $a$. We can always find sides like this if the sides are not all the same length. The shortest $a$ can be is $6$, which allows $c=2$ but then $b$ would have to be $8$, violating the choice of $a$ as a maximal side. If $a \ge 8, (a+c)^2 \ge (a+2)^2 \gt 8a$ and again $b \gt a$ contrary to our selection of $a$. So the only solution is that all sides are $2$.