 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-1/6/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

β = - α

, then both Cartan integers are

- 2

. This opposite-root case satisfies the printed inner-product hypothesis and must be included.

Otherwise reducedness makes the roots nonproportional. Writing the angle as

θ

n_{α, β} n_{β, α} = 4 cos^{2} θ < 4.

Both factors are negative integers, so their product is

1, 2

3

. The full list is therefore

n_{α, β} - 1 - 2 - 3 possible n_{β, α} - 1, - 2, - 3 - 1 or - 2 for opposite roots - 1

The nonproportional pairs correspond respectively to angles

12 0^{\circ}

13 5^{\circ}

and

15 0^{\circ}

; reversing the order of the roots reverses the unequal pairs.

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