Yeah sure.. Newton's Sums..

Algebra Level 5

Let (x1,y1,z1)(x_1, y_1, z_1), (x2,y2,z2)(x_2, y_2, z_2), \dots, (xn,yn,zn)(x_n, y_n, z_n) be the rational solutions to the following system of equations

x+y+z=0x3+y3+z3=18x7+y7+z7=2058x+y+z=0 \\ x^3 + y^3 + z^3 =18 \\ x^7 + y^7 + z^7 =2058

Evaluate:

i=1nxiyizi\sum_{i=1}^{n} x_{i}y_{i}z_{i}

  • The triple (a,b,c)(a, b, c) and (c,a,b)(c, a, b) are considered to be different.
  • This problem is not entirely original.
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