1. The problem asks us to find integers $x$, $y$, and $z$ such that $$x^3 + y^3 + z^3 = 33.$$ 2. This is a famous type of Diophantine equation known as a sum of three cubes problem, which is known to be very challenging for many values of the right-hand side. 3. For some numbers like 33, solutions are not obvious and have been found only recently using extensive computational searches. 4. One known solution for 33 is: $$x = 8866128975287528, \quad y = -8778405442862239, \quad z = -2736111468807040.$$ 5. These are very large integers, showing the complexity of the problem. 6. To verify, you can compute $$x^3 + y^3 + z^3$$ and confirm it equals 33. 7. There is no simpler small integer solution known for this equation. Final answer: $$x=8866128975287528, y=-8778405442862239, z=-2736111468807040.$$