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system x^3 + y^3 = 2, kx + d = y, has no real solutions

Source: New Zealand NZMOC Camp Selection Problems 2014 p7

September 19, 2021
algebrasystem of equationsSystem

Problem Statement

Determine all pairs of real numbers (k,d)(k, d) such that the system of equations {x3+y3=2kx+d=y\begin{cases} x^3 + y^3 = 2 \\ kx + d = y\end{cases} has no solutions (x,y)(x, y) with xx and yy real numbers.
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