35th Vietnamese Mathematical Olympiad 1997 Problems

35th Vietnamese Mathematical Olympiad 1997 Problems

A1.  S is a fixed circle with radius R. P is a fixed point inside the circle with OP = d < R. ABCD is a variable quadrilateral, such that A, B, C, D lie on S, AC intersects BD at P, and AC is perpendicular to BD. Find the maximum and minimum values of the perimeter of ABCD in terms of R and d.

A2.  n > 1 is any integer not divisible by 1997. Put a_m = m + mn/1997 for m = 1, 2, ... , 1996 and b_m = m + 1997m/n for m = 1, 2, ... , n-1. Arrange all the terms a_i, b_j in a single sequence in ascending order. Show that the difference between any two consecutive terms is less than 2.
A3.  How many functions f(n) defined on the positive integers with positive integer values satisfy f(1) = 1 and f(n) f(n+2) = f(n+1)² + 1997 for all n?
B1.  Let k = 3^1/3. Find a polynomial p(x) with rational coefficients and degree as small as possible such that p(k + k²) = 3 + k. Does there exist a polynomial q(x) with integer coefficients such that q(k + k²) = 3 + k?
B2.  Show that for any positive integer n, we can find a positive integer f(n) such that 19^f(n) - 97 is divisible by 2ⁿ.
B3.  Given 75 points in a unit cube, no three collinear, show that we can choose three points which form a triangle with area at most 7/72.

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