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The 35th Vietnam Mathematical Olympiad
1997年第35届越南数学奥林匹克

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 am = m + mn/1997 for m = 1, 2, ... , 1996 and bm = m + 1997m/n for m = 1, 2, ... , n-1. Arrange all the terms ai, bj 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 = 31/3. 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 19f(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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