06/06/2020 (2011: National Sprint Round, Problem 17)

Q: A square has sides of length 4 units each. A stripe of width 1 unit is drawn in the square, centered on the diagonal. The area of this stripe can be expressed in the form a + b√2 square units, where a and b are rational numbers. What is the value of a + b? Express your answer as a number to the nearest tenth.

A: 3.5

Because we know that the strip is centered around the diagonal, that means that AX = AY = CM = CN and that XB = MB = ND = YD. Because the width of the stripe is 1, and XAY = 90 degrees (because we're looking at a square), that means that (AX)² + (AY)² = 1². Because AX = AY, this simplifies to 2(AX)² = 1 --> (AX)² = .5 --> AX = √(2) ÷ 2.

Because AX = √(2) ÷ 2, so XB = 4 - (√(2) ÷ 2), which means MB = 4 - (√(2) ÷ 2). That means the sum of the grey parts is (2)(1/2)(4 - (√(2) ÷ 2))² = 16.5 - 4√(2).

The area of the square is 4² = 16, and the square consists of just white and grey parts, so the white + grey = 16.

That means white + 16.5 - 4√(2) = 16 --> white = 4√(2) - .5 --> a = -.5; b = 4 --> a + b = 4 - .5 = 3.5