 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ia/paper-1/7c/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

(x, y, z)

and

(1, 1, 1)

in the Euclidean inner product on

R^{3}

. It gives

(x + y + z)^{2} \leq 3 (x^{2} + y^{2} + z^{2}) .

After expansion and division by two,

x^{2} + y^{2} + z^{2} \geq x y + yz + z x .

The equality criterion from the introductory proof says

x = y = z

. One can also see both the inequality and equality case from

x^{2} + y^{2} + z^{2} - x y - yz - z x = \frac{1}{2} [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]

 New to topics? Read the docs here!