 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-1/7h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Introduce nonnegative dual variables

y

and

z

maximise subject to t y + rz b_{i} y + c_{i} z \leq a_{i} (1 \leq i \leq n), y, z \geq 0.

Indeed, if

x

and

(y, z)

are feasible, then

t y + rz \leq y \sum_{i} b_{i} x_{i} + z \sum_{i} c_{i} x_{i} = \sum_{i} (b_{i} y + c_{i} z) x_{i} \leq \sum_{i} a_{i} x_{i},

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