IMC 2008 Q19
19. Which of the following is equal to \((1 + x + y)^2 - (1 - x - y)^2\) for all values of \(x\) and \(y\) ?
- A \(4x\)
- B \(2(x^2 + y^2)\)
- C 0
- D \(4xy\)
- E \(4(x + y)\)
Intermediate Mathematical Challenge 2008, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E \(4(x + y)\)
Using ‘the difference of two squares’:
\((1 + x + y)^2 - (1 - x - y)^2 = (1 + x + y + 1 - x - y)(1 + x + y - 1 + x + y) = 2(2x + 2y) = 4(x + y)\).
Solution © UK Mathematics Trust, from the IMC 2008 solutions.