Library / UKMT Intermediate Mathematical Challenge /160 of 267

UKMT Intermediate Mathematical Challenge, 2019

Combinatorics Difficulty 2.0 Junior Find the answer United Kingdom

There are four people, some of whom always tell the truth. The others always lie. The first person said, "An odd number of us always tell the truth". The second person said, "An even number of us always tell the truth". The third person said, "A prime number of us always tell the truth". The fourth person said, "A square number of us always tell the truth". How many of these four people were telling the truth?

(A) $0$ (B) $1$ (C) $2$ (D) $3$ (E) $4$

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.