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1. Given real A, find all solutions (x1, x2, ... , xn) to the n equations (i = 1, 2, ... , n):
xi |xi|- (xi - A) |(xi - A)| = xi+1 |xi+1|, where we take xn+1 to mean x1. (France 1) |
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| 2. Prove that there are infinitely many triples of positive integers (m, n, p) satisfying 4mn - m - n = p2 - 1, but none satisfying 4mn - m - n = p2. (Canada 2) |
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| 3. Find all positive integers n such that n = d62 + d72 - 1, where 1 = d1 < d2 < ... < dk = n are all the positive divisors of n. (USSR 3) | |
| 6. c is a positive integer. The sequence f1, f2, f3, ... is defined by f1 = 1, f2 = c, fn+1 = 2 fn - fn-1 + 2. Show that for each k there is an r such that fk fk+1 = fr. (Canada 3) |
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7. 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?
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9. 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) |
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| 10. Prove that the product of five consecutive positive integers cannot be the square of an integer. (Great Britain 1) |
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| 11. a1, a2, ... , a2n are distinct integers. Find all integers x which satisfy (x - a1)(x - a2) ... (x - a2n) = (-1)n(n!)2. (Canada 1) | |
| 13. 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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15. The angles of the triangle ABC are all < 120o. 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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| 17. If (x1, x2, ... , xn) is a permutation of (1, 2, ... , n) we call the pair (xi, xj) discordant if i < j and xi > xj. 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) | |
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18. 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.
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19. 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 an,1 = 1/n and an,k+1 = an-1,k - an,k. Find the harmonic mean of the 1985th row. (Canada 5) |
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| 20. Find all pairs of positive reals (a, b) with a not 1 such that logab < loga+1(b+1). (USA 2) | |
| Note: This list does not incude the problems used in the Olympiad (4, 5, 8, 12, 14, 16 on the shortlist, which were 5, 1, 3, 2, 4, 6 in the Olympiad). |
Shortlist home
© John Scholes
jscholes@kalva.demon.co.uk
26 Jul 2003
Last corrected/updated 26 Jul 03
(German Federal Republic 5)
(Luxembourg 2)
(USA 5)