Choose distinct <prime numbers> $p,q$ so large that $pq>C(p+q)$, and set
$$
\alpha=\frac1p,
\qquad
\beta=\frac1q.
$$
The fraction $(p+q)/(pq)$ is reduced, because neither $p$ nor $q$ divides $p+q$. The <height of a rational number> therefore gives
$$
H(\alpha+\beta)=pq,
\qquad
H(\alpha)+H(\beta)=p+q.
$$
The choice of $p,q$ proves the required strict inequality.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-166/2/g/solution