The 25th International Mathematical Olympiad Shortlist Problems
1984年第25届国际数学奥林匹克备选题 
 Given real A, find all solutions (x_{1}, x_{2}, ... , x_{n}) to the n equations (i = 1, 2, ... , n):
x_{i} x_{i} (x_{i}  A) (x_{i}  A) = x_{i+1} x_{i+1}, where we take x_{n+1} to mean x_{1}. (France 1)
 Prove that there are infinitely many triples of positive integers (m, n, p) satisfying 4mn  m  n = p^{2}  1, but none satisfying 4mn  m  n = p^{2}. (Canada 2)
 Find all positive integers n such that n = d_{6}^{2} + d_{7}^{2}  1, where 1 = d_{1} < d_{2} < ... < d_{k} = n are all the positive divisors of n. (USSR 3)
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 c is a positive integer. The sequence f_{1}, f_{2}, f_{3}, ... is defined by f_{1} = 1, f_{2} = c, f_{n+1} = 2 f_{n}  f_{n1} + 2. Show that for each k there is an r such that f_{k} f_{k+1} = f_{r}. (Canada 3)
 Can we number the squares of an 8 x 8 board with the numbers 1, 2, ... , 64 so that any four squares with any of the following shapes
have sum = 0 mod 4? Can we do it for the following shapes ?
(German Federal Republic 5)
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 Let a, b, c be positive reals such that √a + √b + √c = √3/2. Show that the equations:
√(y  a) + √(z  a) = 1
√(z  b) + √(x  b) = 1
√(x  c) + √(y  c) = 1
have exactly one solution in reals x, y, z. (Poland 2)
 Prove that the product of five consecutive positive integers cannot be the square of an integer. (Great Britain 1)
 a_{1}, a_{2}, ... , a_{2n} are distinct integers. Find all integers x which satisfy (x  a_{1})(x  a_{2}) ... (x  a_{2n}) = (1)^{n}(n!)^{2}. (Canada 1)
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 A tetrahedron is inscribed in a straight circular cylinder of volume 1. Show that its volume cannot exceed 2/(3π). (Bulgaria 5)
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 The angles of the triangle ABC are all < 120^{o}. Equilateral triangles are constructed on the outside of each side as shown. Show that the three lines AD, BE, CF are concurrent. Suppose they meet at S. Show that SD + SE + SF = 2(SA + SB + SC).
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 If (x_{1}, x_{2}, ... , x_{n}) is a permutation of (1, 2, ... , n) we call the pair (x_{i}, x_{j}) discordant if i < j and x_{i} > x_{j}. Let d(n, k) be the number of permutations of (1, 2, ... , n) with just k discordant pairs. Find d(n, 2) and d(n, 3). (German Federal Republic 3)
 ABC is a triangle. A circles with the radii shown are drawn inside the triangle each touching two sides and the incircle. Find the radius of the incircle.(USA 5)
 The harmonic table is a triangular array:
1
1/2 1/2
1/3 1/6 1/3
1/4 1/12 1/12 1/4
...
where a_{n,1} = 1/n and a_{n,k+1} = a_{n1,k}  a_{n,k}. Find the harmonic mean of the 1985th row. (Canada 5)
 Find all pairs of positive reals (a, b) with a not 1 such that log_{a}b < log_{a+1}(b+1). (USA 2)

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