1. **Problem statement:** We are given the equation $$x^3 + y^3 + (x+y)^3 + 30xy = 2000$$ where $x,y \in \mathbb{Z}$ (integers). We need to find the value of $x+y$. 2. **Recall the identity:** The sum of cubes formula states: $$x^3 + y^3 = (x+y)^3 - 3xy(x+y)$$ 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. **Group terms involving $xy$:** $$ 2(x+y)^3 + xy(-3(x+y) + 30) = 2000 $$ 6. **Let $s = x+y$ and $p = xy$ for simplicity:** $$ 2s^3 + p(-3s + 30) = 2000 $$ 7. **Rewrite:** $$ 2s^3 + p(30 - 3s) = 2000 $$ 8. **Solve for $p$:** $$ p = \frac{2000 - 2s^3}{30 - 3s} $$ 9. **Since $x,y$ are integers, $p=xy$ must be an integer.** Also, $s = x+y$ is an integer. 10. **Check integer values of $s$ such that denominator $30 - 3s \neq 0$ and $p$ is integer:** - Denominator zero when $30 - 3s = 0 \Rightarrow s=10$, exclude $s=10$. - Try integer values near $s=10$ because $2s^3$ grows fast. 11. **Test $s=5$:** $$ p = \frac{2000 - 2(5)^3}{30 - 3(5)} = \frac{2000 - 2(125)}{30 - 15} = \frac{2000 - 250}{15} = \frac{1750}{15} = 116.66... $$ not integer. 12. **Test $s=8$:** $$ p = \frac{2000 - 2(512)}{30 - 24} = \frac{2000 - 1024}{6} = \frac{976}{6} = 162.66... $$ not integer. 13. **Test $s=9$:** $$ p = \frac{2000 - 2(729)}{30 - 27} = \frac{2000 - 1458}{3} = \frac{542}{3} = 180.66... $$ not integer. 14. **Test $s=7$:** $$ p = \frac{2000 - 2(343)}{30 - 21} = \frac{2000 - 686}{9} = \frac{1314}{9} = 146 $$ integer! 15. **Check if $x,y$ with sum $7$ and product $146$ are integers:** Solve quadratic: $$ t^2 - 7t + 146 = 0 $$ Discriminant: $$ \Delta = 7^2 - 4 \times 146 = 49 - 584 = -535 < 0 $$ No real roots, so no integer $x,y$. 16. **Test $s=6$:** $$ p = \frac{2000 - 2(216)}{30 - 18} = \frac{2000 - 432}{12} = \frac{1568}{12} = 130.66... $$ not integer. 17. **Test $s=4$:** $$ p = \frac{2000 - 2(64)}{30 - 12} = \frac{2000 - 128}{18} = \frac{1872}{18} = 104 $$ integer. 18. **Check quadratic for $s=4, p=104$:** $$ t^2 - 4t + 104 = 0 $$ Discriminant: $$ 16 - 416 = -400 < 0 $$ no integer roots. 19. **Test $s=2$:** $$ p = \frac{2000 - 2(8)}{30 - 6} = \frac{2000 - 16}{24} = \frac{1984}{24} = 82.666... $$ no. 20. **Test $s=1$:** $$ p = \frac{2000 - 2(1)}{30 - 3} = \frac{1998}{27} = 74 $$ integer. 21. **Check quadratic for $s=1, p=74$:** $$ t^2 - t + 74 = 0 $$ Discriminant: $$ 1 - 296 = -295 < 0 $$ no integer roots. 22. **Test $s=0$:** $$ p = \frac{2000 - 0}{30 - 0} = \frac{2000}{30} = 66.66... $$ no. 23. **Test $s=3$:** $$ p = \frac{2000 - 2(27)}{30 - 9} = \frac{2000 - 54}{21} = \frac{1946}{21} = 92.66... $$ no. 24. **Test $s=15$:** $$ p = \frac{2000 - 2(3375)}{30 - 45} = \frac{2000 - 6750}{-15} = \frac{-4750}{-15} = 316.66... $$ no. 25. **Test $s=20$:** $$ p = \frac{2000 - 2(8000)}{30 - 60} = \frac{2000 - 16000}{-30} = \frac{-14000}{-30} = 466.66... $$ no. 26. **Test $s=5$ again for completeness:** no integer. 27. **Try $s=10$ is excluded (denominator zero).** 28. **Try $s=12$:** $$ p = \frac{2000 - 2(1728)}{30 - 36} = \frac{2000 - 3456}{-6} = \frac{-1456}{-6} = 242.66... $$ no. 29. **Try $s=13$:** $$ p = \frac{2000 - 2(2197)}{30 - 39} = \frac{2000 - 4394}{-9} = \frac{-2394}{-9} = 266 $$ integer. 30. **Check quadratic for $s=13, p=266$:** $$ t^2 - 13t + 266 = 0 $$ Discriminant: $$ 169 - 1064 = -895 < 0 $$ no integer roots. 31. **Try $s=14$:** $$ p = \frac{2000 - 2(2744)}{30 - 42} = \frac{2000 - 5488}{-12} = \frac{-3488}{-12} = 290.66... $$ no. 32. **Try $s=11$:** $$ p = \frac{2000 - 2(1331)}{30 - 33} = \frac{2000 - 2662}{-3} = \frac{-662}{-3} = 220.66... $$ no. 33. **Try $s= -1$:** $$ p = \frac{2000 - 2(-1)^3}{30 - 3(-1)} = \frac{2000 + 2}{30 + 3} = \frac{2002}{33} = 60.66... $$ no. 34. **Try $s= -2$:** $$ p = \frac{2000 - 2(-8)}{30 - 3(-2)} = \frac{2000 + 16}{30 + 6} = \frac{2016}{36} = 56 $$ integer. 35. **Check quadratic for $s=-2, p=56$:** $$ t^2 + 2t + 56 = 0 $$ Discriminant: $$ 4 - 224 = -220 < 0 $$ no integer roots. 36. **Try $s= -3$:** $$ p = \frac{2000 - 2(-27)}{30 - 3(-3)} = \frac{2000 + 54}{30 + 9} = \frac{2054}{39} = 52.66... $$ no. 37. **Try $s= -4$:** $$ p = \frac{2000 - 2(-64)}{30 - 3(-4)} = \frac{2000 + 128}{30 + 12} = \frac{2128}{42} = 50.66... $$ no. 38. **Try $s= -5$:** $$ p = \frac{2000 - 2(-125)}{30 - 3(-5)} = \frac{2000 + 250}{30 + 15} = \frac{2250}{45} = 50 $$ integer. 39. **Check quadratic for $s=-5, p=50$:** $$ t^2 + 5t + 50 = 0 $$ Discriminant: $$ 25 - 200 = -175 < 0 $$ no integer roots. 40. **Try $s= -6$:** $$ p = \frac{2000 - 2(-216)}{30 - 3(-6)} = \frac{2000 + 432}{30 + 18} = \frac{2432}{48} = 50.66... $$ no. 41. **Try $s= -10$:** $$ p = \frac{2000 - 2(-1000)}{30 - 3(-10)} = \frac{2000 + 2000}{30 + 30} = \frac{4000}{60} = 66.66... $$ no. 42. **Try $s= 20$:** no integer. 43. **Try $s= 0$:** no integer. 44. **Try $s= 1$ to $9$ no integer roots for $p$ with integer roots for quadratic.** 45. **Try $s= 5$ to $9$ no integer roots. 46. **Try $s= 3$ no integer roots. 47. **Try $s= 7$ no integer roots. 48. **Try $s= 4$ no integer roots. 49. **Try $s= 13$ no integer roots. 50. **Try $s= 14$ no integer roots. 51. **Try $s= 15$ no integer roots. 52. **Try $s= 16$:** $$ p = \frac{2000 - 2(4096)}{30 - 48} = \frac{2000 - 8192}{-18} = \frac{-6192}{-18} = 344 $$ integer. 53. **Check quadratic for $s=16, p=344$:** $$ t^2 - 16t + 344 = 0 $$ Discriminant: $$ 256 - 1376 = -1120 < 0 $$ no integer roots. 54. **Try $s= 17$:** $$ p = \frac{2000 - 2(4913)}{30 - 51} = \frac{2000 - 9826}{-21} = \frac{-7826}{-21} = 372.66... $$ no. 55. **Try $s= 18$:** $$ p = \frac{2000 - 2(5832)}{30 - 54} = \frac{2000 - 11664}{-24} = \frac{-9664}{-24} = 402.66... $$ no. 56. **Try $s= 19$:** $$ p = \frac{2000 - 2(6859)}{30 - 57} = \frac{2000 - 13718}{-27} = \frac{-11718}{-27} = 434 $$ integer. 57. **Check quadratic for $s=19, p=434$:** $$ t^2 - 19t + 434 = 0 $$ Discriminant: $$ 361 - 1736 = -1375 < 0 $$ no integer roots. 58. **Try $s= 21$:** $$ p = \frac{2000 - 2(9261)}{30 - 63} = \frac{2000 - 18522}{-33} = \frac{-16522}{-33} = 500.66... $$ no. 59. **Try $s= 22$:** $$ p = \frac{2000 - 2(10648)}{30 - 66} = \frac{2000 - 21296}{-36} = \frac{-19296}{-36} = 536 $$ integer. 60. **Check quadratic for $s=22, p=536$:** $$ t^2 - 22t + 536 = 0 $$ Discriminant: $$ 484 - 2144 = -1660 < 0 $$ no integer roots. 61. **Try $s= 23$:** $$ p = \frac{2000 - 2(12167)}{30 - 69} = \frac{2000 - 24334}{-39} = \frac{-22334}{-39} = 572.66... $$ no. 62. **Try $s= 24$:** $$ p = \frac{2000 - 2(13824)}{30 - 72} = \frac{2000 - 27648}{-42} = \frac{-25648}{-42} = 610.66... $$ no. 63. **Try $s= 25$:** $$ p = \frac{2000 - 2(15625)}{30 - 75} = \frac{2000 - 31250}{-45} = \frac{-29250}{-45} = 650 $$ integer. 64. **Check quadratic for $s=25, p=650$:** $$ t^2 - 25t + 650 = 0 $$ Discriminant: $$ 625 - 2600 = -1975 < 0 $$ no integer roots. 65. **Try $s= 26$:** $$ p = \frac{2000 - 2(17576)}{30 - 78} = \frac{2000 - 35152}{-48} = \frac{-33152}{-48} = 690.66... $$ no. 66. **Try $s= 27$:** $$ p = \frac{2000 - 2(19683)}{30 - 81} = \frac{2000 - 39366}{-51} = \frac{-37366}{-51} = 732.66... $$ no. 67. **Try $s= 28$:** $$ p = \frac{2000 - 2(21952)}{30 - 84} = \frac{2000 - 43904}{-54} = \frac{-41904}{-54} = 776 $$ integer. 68. **Check quadratic for $s=28, p=776$:** $$ t^2 - 28t + 776 = 0 $$ Discriminant: $$ 784 - 3104 = -2320 < 0 $$ no integer roots. 69. **Try $s= 29$:** $$ p = \frac{2000 - 2(24389)}{30 - 87} = \frac{2000 - 48778}{-57} = \frac{-46778}{-57} = 820.66... $$ no. 70. **Try $s= 30$:** $$ p = \frac{2000 - 2(27000)}{30 - 90} = \frac{2000 - 54000}{-60} = \frac{-52000}{-60} = 866.66... $$ no. 71. **Try $s= 31$:** $$ p = \frac{2000 - 2(29791)}{30 - 93} = \frac{2000 - 59582}{-63} = \frac{-57582}{-63} = 913.66... $$ no. 72. **Try $s= 32$:** $$ p = \frac{2000 - 2(32768)}{30 - 96} = \frac{2000 - 65536}{-66} = \frac{-63536}{-66} = 962 $$ integer. 73. **Check quadratic for $s=32, p=962$:** $$ t^2 - 32t + 962 = 0 $$ Discriminant: $$ 1024 - 3848 = -2824 < 0 $$ no integer roots. 74. **Try $s= 33$:** $$ p = \frac{2000 - 2(35937)}{30 - 99} = \frac{2000 - 71874}{-69} = \frac{-69874}{-69} = 1012.66... $$ no. 75. **Try $s= 34$:** $$ p = \frac{2000 - 2(39304)}{30 - 102} = \frac{2000 - 78608}{-72} = \frac{-76608}{-72} = 1064 $$ integer. 76. **Check quadratic for $s=34, p=1064$:** $$ t^2 - 34t + 1064 = 0 $$ Discriminant: $$ 1156 - 4256 = -3100 < 0 $$ no integer roots. 77. **Try $s= 35$:** $$ p = \frac{2000 - 2(42875)}{30 - 105} = \frac{2000 - 85750}{-75} = \frac{-83750}{-75} = 1116.66... $$ no. 78. **Try $s= 36$:** $$ p = \frac{2000 - 2(46656)}{30 - 108} = \frac{2000 - 93312}{-78} = \frac{-91312}{-78} = 1171.18... $$ no. 79. **Try $s= 37$:** $$ p = \frac{2000 - 2(50653)}{30 - 111} = \frac{2000 - 101306}{-81} = \frac{-99206}{-81} = 1225.6... $$ no. 80. **Try $s= 38$:** $$ p = \frac{2000 - 2(54872)}{30 - 114} = \frac{2000 - 109744}{-84} = \frac{-107744}{-84} = 1282.19... $$ no. 81. **Try $s= 39$:** $$ p = \frac{2000 - 2(59319)}{30 - 117} = \frac{2000 - 118638}{-87} = \frac{-116638}{-87} = 1340.44... $$ no. 82. **Try $s= 40$:** $$ p = \frac{2000 - 2(64000)}{30 - 120} = \frac{2000 - 128000}{-90} = \frac{-126000}{-90} = 1400 $$ integer. 83. **Check quadratic for $s=40, p=1400$:** $$ t^2 - 40t + 1400 = 0 $$ Discriminant: $$ 1600 - 5600 = -4000 < 0 $$ no integer roots. 84. **Summary:** For all integer $s$ tested, $p$ integer but quadratic has no integer roots. 85. **Try $s= 3$ and $p= 92.66$ no integer roots. 86. **Try $s= 1$ and $p= 74$ no integer roots. 87. **Try $s= 0$ no integer roots. 88. **Try $s= 10$ denominator zero. 89. **Try $s= 5$ no integer roots. 90. **Try $s= 7$ no integer roots. 91. **Try $s= 4$ no integer roots. 92. **Try $s= 6$ no integer roots. 93. **Try $s= 8$ no integer roots. 94. **Try $s= 9$ no integer roots. 95. **Try $s= -1$ no integer roots. 96. **Try $s= -2$ no integer roots. 97. **Try $s= -3$ no integer roots. 98. **Try $s= -4$ no integer roots. 99. **Try $s= -5$ no integer roots. 100. **Try $s= -6$ no integer roots. 101. **Try $s= -7$:** $$ p = \frac{2000 - 2(-343)}{30 - 3(-7)} = \frac{2000 + 686}{30 + 21} = \frac{2686}{51} = 52.66... $$ no. 102. **Try $s= -8$:** $$ p = \frac{2000 - 2(-512)}{30 - 3(-8)} = \frac{2000 + 1024}{30 + 24} = \frac{3024}{54} = 56 $$ integer. 103. **Check quadratic for $s=-8, p=56$:** $$ t^2 + 8t + 56 = 0 $$ Discriminant: $$ 64 - 224 = -160 < 0 $$ no integer roots. 104. **Try $s= -9$:** $$ p = \frac{2000 - 2(-729)}{30 - 3(-9)} = \frac{2000 + 1458}{30 + 27} = \frac{3458}{57} = 60.66... $$ no. 105. **Try $s= -10$:** $$ p = \frac{2000 - 2(-1000)}{30 - 3(-10)} = \frac{2000 + 2000}{30 + 30} = \frac{4000}{60} = 66.66... $$ no. 106. **Try $s= -11$:** $$ p = \frac{2000 - 2(-1331)}{30 - 3(-11)} = \frac{2000 + 2662}{30 + 33} = \frac{4662}{63} = 74 $$ integer. 107. **Check quadratic for $s=-11, p=74$:** $$ t^2 + 11t + 74 = 0 $$ Discriminant: $$ 121 - 296 = -175 < 0 $$ no integer roots. 108. **Try $s= -12$:** $$ p = \frac{2000 - 2(-1728)}{30 - 3(-12)} = \frac{2000 + 3456}{30 + 36} = \frac{5456}{66} = 82.66... $$ no. 109. **Try $s= -13$:** $$ p = \frac{2000 - 2(-2197)}{30 - 3(-13)} = \frac{2000 + 4394}{30 + 39} = \frac{6394}{69} = 92.66... $$ no. 110. **Try $s= -14$:** $$ p = \frac{2000 - 2(-2744)}{30 - 3(-14)} = \frac{2000 + 5488}{30 + 42} = \frac{7488}{72} = 104 $$ integer. 111. **Check quadratic for $s=-14, p=104$:** $$ t^2 + 14t + 104 = 0 $$ Discriminant: $$ 196 - 416 = -220 < 0 $$ no integer roots. 112. **Try $s= -15$:** $$ p = \frac{2000 - 2(-3375)}{30 - 3(-15)} = \frac{2000 + 6750}{30 + 45} = \frac{8750}{75} = 116.66... $$ no. 113. **Try $s= -16$:** $$ p = \frac{2000 - 2(-4096)}{30 - 3(-16)} = \frac{2000 + 8192}{30 + 48} = \frac{10192}{78} = 130.66... $$ no. 114. **Try $s= -17$:** $$ p = \frac{2000 - 2(-4913)}{30 - 3(-17)} = \frac{2000 + 9826}{30 + 51} = \frac{11826}{81} = 146 $$ integer. 115. **Check quadratic for $s=-17, p=146$:** $$ t^2 + 17t + 146 = 0 $$ Discriminant: $$ 289 - 584 = -295 < 0 $$ no integer roots. 116. **Try $s= -18$:** $$ p = \frac{2000 - 2(-5832)}{30 - 3(-18)} = \frac{2000 + 11664}{30 + 54} = \frac{13664}{84} = 162.66... $$ no. 117. **Try $s= -19$:** $$ p = \frac{2000 - 2(-6859)}{30 - 3(-19)} = \frac{2000 + 13718}{30 + 57} = \frac{15718}{87} = 180.66... $$ no. 118. **Try $s= -20$:** $$ p = \frac{2000 - 2(-8000)}{30 - 3(-20)} = \frac{2000 + 16000}{30 + 60} = \frac{18000}{90} = 200 $$ integer. 119. **Check quadratic for $s=-20, p=200$:** $$ t^2 + 20t + 200 = 0 $$ Discriminant: $$ 400 - 800 = -400 < 0 $$ no integer roots. 120. **Conclusion:** No integer $x,y$ satisfy the equation with $x+y=s$ and $xy=p$ integer and quadratic having integer roots. 121. **Re-examine original equation:** Try $x=y$: $$ 2x^3 + (2x)^3 + 30x^2 = 2000 $$ $$ 2x^3 + 8x^3 + 30x^2 = 2000 $$ $$ 10x^3 + 30x^2 = 2000 $$ $$ 10x^3 + 30x^2 - 2000 = 0 $$ Divide by 10: $$ x^3 + 3x^2 - 200 = 0 $$ Try integer roots dividing 200: $x=5$: $$ 125 + 75 - 200 = 0 $$ true. 122. **So $x=y=5$, then $x+y=10$.** 123. **Check denominator for $s=10$:** $$ 30 - 3(10) = 0 $$ denominator zero, but original substitution was invalid for $s=10$. 124. **Check original equation with $x=y=5$:** $$ 5^3 + 5^3 + (5+5)^3 + 30 \times 5 \times 5 = 125 + 125 + 1000 + 750 = 2000 $$ correct. 125. **Answer:** $$ \boxed{10} $$