15th Asian Pacific Mathematics Olympiad 2003 Problems

15th Asian Pacific Mathematics Olympiad 2003 Problems 1.  The polynomial a8x8 +a7x7 + ... + a0 has a8 = 1, a7 = -4, a6 = 7 and all its roots positive and real. Find the possible values for a0. 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? A2.  Find an n in the range 100,

8th Asian Pacific Mathematics Olympiad 1996 Problems

8th Asian Pacific Mathematics Olympiad 1996 ProblemsA1.  ABCD is a fixed rhombus. P lies on AB and Q on BC, so that PQ is perpendicular to BD. Similarly P' lies on AD and Q' on CD, so that P'Q' is perpendicular to BD. The distance between PQ and P'Q' is more than BD/2. Show that the perimeter of the hexagon APQCQ'P' depends only on the distance between PQ and P'Q'.

7th Asian Pacific Mathematics Olympiad 1995 Problems

7th Asian Pacific Mathematics Olympiad 1995 ProblemsA1.  Find all real sequences x1, x2, ... , x1995 which satisfy 2√(xn - n + 1) ≥ xn+1 - n + 1 for n = 1, 2, ... , 1994, and 2√(x1995 - 1994) ≥ x1 + 1. A2. 

6th Asian Pacific Mathematics Olympiad 1994 Problems

6th Asian Pacific Mathematics Olympiad 1994 ProblemsA1.  Find all real-valued functions f on the reals such that (1) f(1) = 1, (2) f(-1) = -1, (3) f(x) ≤ f(0) for 0 < x < 1, (4) f(x + y) ≥ f(x) + f(y) for all x, y, (5) f(x + y) ≤ f(x) + f(y) + 1 for all x, y. A2.

4th Asian Pacific Mathematics Olympiad 1992 Problems

4th Asian Pacific Mathematics Olympiad 1992 Problems A1.  A triangle has sides a, b, c. Construct another triangle sides (-a + b + c)/2, (a - b + c)/2, (a + b - c)/2. For which triangles can this process be repeated arbitrarily many times?