 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iv/1/c/solution

The value

f (a, b, c)

is twice the signed area of the triangle traversed in that order; it is positive for counterclockwise orientation. Choose points on the parabola

p_{i} = (i, i^{2}), 1 \leq i \leq n .

For

i < j < k

, subtracting the first row in the determinant and factoring gives

f (p_{i}, p_{j}, p_{k}) = (j - i) (k^{2} - i^{2}) - (k - i) (j^{2} - i^{2}) = (j - i) (k - i) (k - j) > 0.

Each of the three factors is positive, proving the required property for every integer

n \geq 3

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