Problem 4

Find all sets of four real numbers

x_1,x_2,x_3,x_4

such that the sum of any one number and the product of the other three is equal to

2

Step 1 of 5: Exclude a zero coordinate

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x_i=0\\Longrightarrow x_j=2\\ (j\ne i)\\Longrightarrow x_jx_kx_l=8\ne2,\\quad\text{so all }x_i\ne0.

Detailed analysis

If one number, say $x_1$ , were zero, the equation indexed by $x_2$ would force $x_2=2$ , and similarly $x_3=x_4=2$ . But then the equation indexed by $x_1$ has product $x_2x_3x_4=8$ , not $2$ . Thus division by every $x_i$ is legitimate.