 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-4/7f/d/solution

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

y_{i} = {x_{i}, 0, s_{i} > 0, s_{i} = 0, z_{i} = x_{i} - y_{i} .

Then

x = y + z

, the support of

y

lies in

P (s)

, and the support of

z

is disjoint from

P (s)

. Thus

y ⪯ s

and

z ⊥ s

s_{i} > 0

z_{i} = 0

, hence

y_{i} = x_{i}

. If

s_{i} = 0

, the condition

y ⪯ s

y_{i} = 0

, hence

z_{i} = x_{i}

. Every coordinate is therefore forced, proving uniqueness.

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