Find all integers $x$ such that $x^{3}+(x+1)^{3}+(x+2)^{3}=(x+3)^{3}$.
Let's develop both sides of the equality:

$$
x^{3}+x^{3}+3 x^{2}+3 x+1+x^{3}+6 x^{2}+12 x+8=x^{3}+9 x^{2}+27 x+27
$$

, so

$x^{3}-6 x-9=0$. We deduce that $x$ divides 9, so $x$ is $-9,-3,-1,1,3$ or 9. Of all these possibilities, only 3 is correct. The solution to this equation is therefore 3.