 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2010/f/f2/c/2/solution

Let

S = \sum_{i} p_{i} a_{i} > 0

and set

x_{i} = a_{i} / S

. Then

x_{i} > 0

and

\sum_{i} p_{i} x_{i} = 1

, so part (1) gives

1 \geq \prod_{i} x_{i}^{p_{i}} = \frac{\prod _{i} a _{i}^{p_{i}}}{S ^{\sum_{i} p_{i}}} = \frac{\prod _{i} a _{i}^{p_{i}}}{S} .

i \prod a_{i}^{p_{i}} \leq i \sum p_{i} a_{i} .

Equality holds exactly when all

a_{i}

are equal, by the equality case in part (1).

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