IMC 2020 Q5
5. Four of the following coordinate pairs are the corners of a square.
Which is the odd one out?
- A (4, 1)
- B (2, 4)
- C (5, 6)
- D (3, 5)
- E (7, 3)
Intermediate Mathematical Challenge 2020, Question 5. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D (3, 5)
First note that \(AB = \sqrt{13}\), \(AC = \sqrt{26}\), \(AD = \sqrt{17}\) and \(AE = \sqrt{13}\). Thus \(AB = AE\).
Note also that \(BC = \sqrt{13}\) and \(CE = \sqrt{13}\). Therefore \(ABCE\) is a rhombus.
Finally, \(BE = \sqrt{26}\), so \(ABCE\) is a rhombus with two equal diagonals and hence is a square.
So (3,5) is the odd point out.
Solution © UK Mathematics Trust, from the IMC 2020 solutions. Longer versions with investigations: Solutions and investigations (PDF).