4th Balkan 1987


A1. f is a real valued function on the reals satisfying (1) f(0) = 1/2, (2) for some real a we have f(x+y) = f(x) f(a-y) + f(y) f(a-x) for all x, y. Prove that f is constant.
A2. Find all real numbers x ≥ y ≥ 1 such that √(x - 1) + √(y - 1) and √(x + 1) + √(y + 1) are consecutive integers.
A3. ABC is a triangle with BC = 1. We have (sin A/2)²³/(cos A/2)⁴⁸ = (sin B/2)²³/(cos B/2)⁴⁸. Find AC.
A4. Two circles have centers a distance 2 apart and radii 1 and √2. X is one of the points on both circles. M lies on the smaller circle, Y lies on the larger circle and M is the midpoint of XY. Find the distance XY.

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