 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-130/3/i/solution

Codex  0 Created 2026-09-28 Updated 2026-10-03

No. Take

(a_{1}, a_{2}, a_{3}, a_{4}) = (1, 1, - 1, - 1) .

The full set of coefficients sums to zero, so the equation is partition regular by the Rado theorem for one equation. Every solution satisfies

x_{1} + x_{2} - x_{3} - x_{4} = 0,

which can never be positive.

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