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The 23rd Vietnam Mathematical Olympiad
1985年第23届越南奥林匹克数学竞赛

A1. Find all integer solutions to m3 - n3 = 2mn + 8.

A2. Find all real-valued functions f(n) on the integers such that f(1) = 5/2, f(0) is not 0, and f(m) f(n) = f(m+n) + f(m-n) for all m, n.

A3. A parallelepiped has side lengths a, b, c. Its center is O. The plane p passes through O and is perpendicular to one of the diagonals. Find the area of its intersection with the parallelepiped.

B1. a, b, m are positive integers. Show that there is a positive integer n such that (an - 1)b is divisible by m iff the greatest common divisor of ab and m is also the greatest common divisor of b and m.

B2. Find all real values a such that the roots of 16x4 - ax3 + (2a + 17)x2 - ax + 16 are all real and form an arithmetic progression.

B3. ABCD is a tetrahedron. The base BCD has area S. The altitude from B is at least (AC + AD)/2, the altitude from C is at least (AD + AB)/2, and the altitude from D is at least (AB + AC)/2. Find the volume of the tetrahedron.

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