Problem 1 - Entrance Test

Let I = ∫₀^(π/2) (sin(x) / (sin(x) + cos(x))) dx. Using the property ∫₀^a f(x) dx = ∫₀^a f(a-x) dx, we have I = ∫₀^(π/2) (sin(π/2 - x) / (sin(π/2 - x) + cos(π/2 - x))) dx = ∫₀^(π/2) (cos(x) / (cos(x) + sin(x))) dx. Adding the two equations, 2I = ∫₀^(π/2) ((sin(x) + cos(x)) / (sin(x) + cos(x))) dx = ∫₀^(π/2) 1 dx = [x]₀^(π/2) = π/2. Therefore, I = π/4.