Library / UKMT Senior Kangaroo /106 of 139

UKMT Senior Kangaroo, 2023

Combinatorics Difficulty 3.0 AMC 10/12 Prove it United Kingdom

Stephen's calculator displays only one digit, as shown in the diagram. Unfortunately, the calculator is broken. Each time he switches it on, each of the seven bars will either illuminate (show up) or not, with probability $0.5$ . The resultant display correctly shows one of the ten digits $0$ - $9$ with probability $\frac{a}{b}$ . Given that $\frac{a}{b}$ is written in its lowest terms, what is the value of $9a + 2b$ ?

Figure for this problem

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.