## Task 1 - 181231

Determine all polynomials $f(x)$ with real coefficients that satisfy the equation $f(x+1)-f(x)=x+1$ for all real $x$.
We choose the approach

$$
f(x)=a x^{2}+b x+c
$$

Then we have

$$
f(x+1)=a x^{2}+2 a x+a+b x+b+c
$$

By substituting into the problem statement, we get:

$$
2 a x+a+b=x+1
$$

from which $a=\frac{1}{2}$ and $b=\frac{1}{2}$ follow. Therefore, all functions

$$
f(x)=\frac{1}{2}\left(x^{2}+x\right)+c
$$

satisfy the equation.