This question comes from a math competition I had a few days ago:
Evaluate $$\lceil{\frac{fs}{f+s}}\rceil$$ if $$1+\sum_{k=1}^{2019}k(k^2+k+1) = f! - s!$$
What I noticed first is that you can distribute the $k$:
$$1+\sum_{k=1}^{2019}(k^3+k^2+k) = f! - s!$$ Then you can evaluate each term in the sum individually.
$$1+\sum_{k=1}^{2019}k^3+\sum_{k=1}^{2019}k^2+\ \sum_{k=1}^{2019}k = f! - s!$$
Then by using the formula of sum of cubes, squares, and k to n = 2019 we can get an expression (I didn't evaluate the sum). Then by doing this, I thought to myself that I can probably express the expression in terms of factorials. But to this time I still haven't found a way. Any hint or answer will help. thanks