IMC 2019 Q14
14. 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
Intermediate Mathematical Challenge 2019, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 2
Since no integer is both odd and even, exactly one of the first two speakers was telling the truth. No square is also prime; so at most one of the third and fourth speakers was telling the truth. Hence the number telling the truth must be one or two. If there was just one telling the truth then the statements of speakers 1 and 4 would both be true - which is a contradiction. If there were two telling the truth, then speakers 2 and 3 would be the ones telling the truth; and this is indeed two speakers.
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).