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The 4th Balkan Mathematical Olympiad
1987年第4届巴尔干地区数学奥林匹克
  1. 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.
  2. Find all real numbers x ≥ y ≥ 1 such that √(x - 1) + √(y - 1) and √(x + 1) + √(y + 1) are consecutive integers.
  3. 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.
  4. 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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