4th Austrian-Polish Mathematics Competition 1981 Problems

4th Austrian-Polish Mathematics Competition 1981 Problems1.  Find the smallest n for which we can find 15 distinct elements a1, a2, ... , a15 of {16, 17, ... , n} such that ak is a multiple of k. 2.  The rational

3rd Austrian-Polish Mathematics Competition 1980 Problems

3rd Austrian-Polish Mathematics Competition 1980 Problems1.  A, B, C are infinite arithmetic progressions of integers. {1, 2, 3, 4, 5, 6, 7, 8} is a subset of their union. Show that 1980 also belongs to their union.

2nd Austrian-Polish Mathematics Competition 1979 Problems

2nd Austrian-Polish Mathematics Competition 1979 Problems1.  ABCD is a square. E is any point on AB. F is the point on BC such that BF = BE. The perpendicular from B meets EF at G. Show that ∠DGF = 90o.2.  Find all

1st Austrian-Polish Mathematics Competition 1978 Problems

1st Austrian-Polish Mathematics Competition 1978 Problems1.  Find all real-valued functions f on the positive reals which satisfy f(x + y) = f(x2 + y2) for all positive x, y. 2.  A parallelogram has its vertices on the boundary of a regular hexagon and its center at the center of the hexagon. Show that its area is at most 2/3 the area of the hexagon. 2.  A unit square

14th Asian Pacific Mathematics Olympiad 2002 Problems

14th Asian Pacific Mathematics Olympiad 2002 ProblemsA1.  xi are non-negative integers. Prove that x1! x2! ... xn! ≥ ( [(x1 + ... + xn)/n] ! )n (where [y] denotes the largest integer not exceeding y). When do you have equality?