Estonian Mathematical Olympiad

The given equation is equivalent to $$(11x^5-31x)+(33y-13y^5)=2024.$$ The fifth power of any positive integer ends with the same digit as the number itself. Therefore, $11x^5$ ends with the same digit as $11x$, which in turn ends with the same digit as $x$. Since $31x$ also ends with the same digit, the difference $11x^5-31x$ ends with zero. Similarly, the difference $33y-13y^5$ ends with zero. However, the sum of numbers ending in zero cannot end in four. Therefore, there are no integers that satisfy the given equation.