IMC 2020 Q24
24. The positive integers \(m\) and \(n\) are such that \(10 \times 2^m = 2^n + 2^{n+2}\).
What is the difference between \(m\) and \(n\)?
- A 1
- B 2
- C 3
- D 4
- E 5
Intermediate Mathematical Challenge 2020, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 1
We are given that \(10 \times 2^m = 2^n + 2^{n+2}\). Therefore \(5 \times 2 \times 2^m = 2^n(1 + 2^2)\).
Hence \(5 \times 2^{m+1} = 2^n \times 5\). So \(m + 1 = n\). Therefore the difference between \(m\) and \(n\) is 1.
Solution © UK Mathematics Trust, from the IMC 2020 solutions. Longer versions with investigations: Solutions and investigations (PDF).