IMC 2009 Q24
24. What is the largest number of the following statements that can be true at the same time?
- A 1
- B 2
- C 3
- D 4
- E 5
Intermediate Mathematical Challenge 2009, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 3
Reading from the left, we number the statements I, II, III, IV and V.
Statement I is true if and only if \(-1 < x < 1\); statement II is true if \(x > 1\) or if \(x < -1\).
By considering the graph of \(y = x - x^2\), which intersects the \(x\)-axis at (0, 0) and (1, 0) and has a maximum at \((\tfrac{1}{2}, \tfrac{1}{4})\), it may be seen that statement V is true if and only if \(0 < x < 1\).
We see from the table below that a maximum of three statements may be true at any one time.
Solution © UK Mathematics Trust, from the IMC 2009 solutions.