Germany | Can you solve this? | Math Olympiad 

Interesting 🎉

Muy interesante 😅

Nice! I finished the same way as you but via a longer journey. I defined f(x) as (x^4+x^2+1)/(x^3+x^2+x) and tried some integers ... I noticed that f(n) appeared to be always equal to (n-1) + 1/n ... but unable to see why this should only be true for integers I conjectured that this was in fact an identity for all x. So this led me back to seeing that (x^2+x+1) must be a factor of the numerator in f(x) above. From there I found(proved) by long division that this was the case and along the way noticed the clever diversionary trick using difference of two squares (as you did). Very interesting patterns that I have not seen before.