1. **Stating the problem:** We need to solve the equation $$x^3 + y^3 + (x+y)^3 + 30xy = 2000$$ where $x, y \in \mathbb{Z}$ (integers). 2. **Recall the identity:** The sum of cubes formula for two variables is: $$x^3 + y^3 = (x+y)^3 - 3xy(x+y)$$ 3. **Rewrite the equation using the identity:** Substitute $x^3 + y^3$: $$ (x+y)^3 - 3xy(x+y) + (x+y)^3 + 30xy = 2000 $$ 4. **Combine like terms:** $$ 2(x+y)^3 - 3xy(x+y) + 30xy = 2000 $$ 5. **Let $s = x+y$ and $p = xy$ for simplicity:** The equation becomes: $$ 2s^3 - 3ps + 30p = 2000 $$ 6. **Rearrange to isolate $p$:** $$ -3ps + 30p = 2000 - 2s^3 $$ $$ p(-3s + 30) = 2000 - 2s^3 $$ $$ p = \frac{2000 - 2s^3}{-3s + 30} = \frac{2000 - 2s^3}{30 - 3s} $$ 7. **Since $x,y$ are integers, $p=xy$ must be integer.** We need integer $s$ such that $p$ is integer. 8. **Check integer values of $s$ that make denominator nonzero and $p$ integer:** Denominator $30 - 3s \neq 0 \Rightarrow s \neq 10$. Try values near $s=10$: - For $s=5$: $$ p = \frac{2000 - 2(125)}{30 - 15} = \frac{2000 - 250}{15} = \frac{1750}{15} = 116.66... $$ not integer. - For $s=8$: $$ p = \frac{2000 - 2(512)}{30 - 24} = \frac{2000 - 1024}{6} = \frac{976}{6} = 162.66... $$ not integer. - For $s=4$: $$ p = \frac{2000 - 2(64)}{30 - 12} = \frac{2000 - 128}{18} = \frac{1872}{18} = 104 $$ integer. 9. **For $s=4$, $p=104$.** Solve quadratic for $x,y$: $$ t^2 - st + p = 0 \Rightarrow t^2 - 4t + 104 = 0 $$ Discriminant: $$ \Delta = 16 - 4 \times 104 = 16 - 416 = -400 < 0 $$ No real (and thus no integer) solutions. 10. **Try $s=2$:** $$ p = \frac{2000 - 2(8)}{30 - 6} = \frac{2000 - 16}{24} = \frac{1984}{24} = 82.666... $$ no. 11. **Try $s=1$:** $$ p = \frac{2000 - 2(1)}{30 - 3} = \frac{1998}{27} = 74 $$ integer. Quadratic: $$ t^2 - t + 74 = 0 $$ Discriminant: $$ 1 - 296 = -295 < 0 $$ no integer solutions. 12. **Try $s=0$:** $$ p = \frac{2000 - 0}{30 - 0} = \frac{2000}{30} = 66.66... $$ no. 13. **Try $s=6$:** $$ p = \frac{2000 - 2(216)}{30 - 18} = \frac{2000 - 432}{12} = \frac{1568}{12} = 130.66... $$ no. 14. **Try $s=3$:** $$ p = \frac{2000 - 2(27)}{30 - 9} = \frac{2000 - 54}{21} = \frac{1946}{21} = 92.66... $$ no. 15. **Try $s=9$:** $$ p = \frac{2000 - 2(729)}{30 - 27} = \frac{2000 - 1458}{3} = \frac{542}{3} = 180.66... $$ no. 16. **Try $s=7$:** $$ p = \frac{2000 - 2(343)}{30 - 21} = \frac{2000 - 686}{9} = \frac{1314}{9} = 146 $$ integer. Quadratic: $$ t^2 - 7t + 146 = 0 $$ Discriminant: $$ 49 - 584 = -535 < 0 $$ no integer solutions. 17. **Try $s=10$ is invalid (denominator zero).** 18. **Try $s=15$:** $$ p = \frac{2000 - 2(3375)}{30 - 45} = \frac{2000 - 6750}{-15} = \frac{-4750}{-15} = 316.66... $$ no. 19. **Try $s=20$:** $$ p = \frac{2000 - 2(8000)}{30 - 60} = \frac{2000 - 16000}{-30} = \frac{-14000}{-30} = 466.66... $$ no. 20. **Try $s=-2$:** $$ p = \frac{2000 - 2(-8)}{30 - (-6)} = \frac{2000 + 16}{36} = \frac{2016}{36} = 56 $$ integer. Quadratic: $$ t^2 + 2t + 56 = 0 $$ Discriminant: $$ 4 - 224 = -220 < 0 $$ no integer solutions. 21. **Try $s=-4$:** $$ p = \frac{2000 - 2(-64)}{30 - (-12)} = \frac{2000 + 128}{42} = \frac{2128}{42} = 50.66... $$ no. 22. **Try $s=-5$:** $$ p = \frac{2000 - 2(-125)}{30 - (-15)} = \frac{2000 + 250}{45} = \frac{2250}{45} = 50 $$ integer. Quadratic: $$ t^2 + 5t + 50 = 0 $$ Discriminant: $$ 25 - 200 = -175 < 0 $$ no integer solutions. 23. **Try $s=-10$:** $$ p = \frac{2000 - 2(-1000)}{30 - (-30)} = \frac{2000 + 2000}{60} = \frac{4000}{60} = 66.66... $$ no. 24. **Conclusion:** No integer pairs $(x,y)$ satisfy the equation because for all integer $s$ tested, either $p$ is not integer or the quadratic has no integer roots. **Final answer:** There are no integer solutions $(x,y)$ to the equation $$x^3 + y^3 + (x+y)^3 + 30xy = 2000$$.