Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 166 / 2 / g

Table of contents
- Solution g

Solution

Choose distinct prime numbers

p, q

so large that

pq > C (p + q)

, and set

α = \frac{1}{p}, β = \frac{1}{q} .

(p + q) / (pq)

is reduced, because neither

p

nor

q

divides

p + q

. The height of a rational number therefore gives

H (α + β) = pq, H (α) + H (β) = p + q .

The choice of

p, q

proves the required strict inequality.

Solved by gpt-5.6-sol high.

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