IMO预选题1984(shortlist)

7. Can we number the squares of an 8 8 board with the numbers 1, 2, ... ,
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non-negative real numbers satisfying x + y + z = 1.
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12. Find one pair of positive integers a, b such that ab(a + b) is not
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3 . Show that 2
x c y c 1
13. A tetrahedron is inscribed in a straight circular cylinder of volume 1. Show that its volume cannot exceed 2/(3 ). (Bulgaria 5)
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(Canada 3)
64 so that any four squares with any of the following shapes have sum = 0 (mod 4)?
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6. c is a positive integer. The sequence f1, f2, f3, ... is defined by f1 = 1, f2 =
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with one of a finite number of colors. Given a point X in the plane, the circle
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9. Let a, b, c be positive reals such that
a b c
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the equations:
5. Prove that 0 yz + zx + xy - 2xyz 7/27, where x, y and z are
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c, f n+1 = 2fn - f n-1 + 2. Show that for each k there is an r such that f kfk+1 = f r.
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). (Luxembourg 2)
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)
4. Let d be the sum of the lengths of all the diagonals of a plane convex polygon with n > 3 vertices. Let p be its perimeter. Prove that: n - 3 < 2d/p < [n/2][(n + 1)/2] - 2, where [x] denotes the greatest integer not exceeding x.
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Can we do it for the following shapes? (German Federal Republic 5)
8. Given points O and A in the plane. Every point in the plane is colored
C(X) has center O and radius OX + ( AOX)/OX, where AOX is measured in
1/12 1/12 1/4
where a n,1 = 1/n and a n,k+1 = a n-1,k - a n,k. Find the harmonic mean of the 1985th row. (Canada 5)
20. Find all pairs of positive reals (a, b) with a not 1 such that logab <
25th IMO 1984 shortlisted problems
1. Given real A, find all solutions (x1, x 2, ... , x n) to the n equations (i = 1, 2, ... , n): x i|xi| - (xi - A)|(xi - A)| = x i+1|xi+1|, where we take x n+1 to mean x 1. (France 1)
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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)
16. Let a, b, c, d be odd integers such that 0 < a < b < c < d and ad = bc. Prove that if a + d = 2k and b + c = 2m for some integers k and m, then a = 1.
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that its color appears on the circumference of the circle C(X).
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radians in the range [0, 2 ). Prove that we can find a point X, not on OA, such
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17. If (x1, x2, ... , xn) is a permutation of (1, 2, ... , n) we call the pair (xi, xj)
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19. The harmonic table is a triangular array: 1 1/2 1/3 1/4 ... 1/2 1/6 1/3
y a z a 1,
z b x b 1 and
have exactly one solution in reals x, y, z. (Poland 2)
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the square of an integer. (Great Britain 1)
a1)(x - a2) ... (x - a2n) = (-1)n(n!)2. (Canada 1)
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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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)
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18. ABC is a triangle. A

divisible by 7, but (a + b)7 - a7 - b7 is divisible by 77.
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10. Prove that the product of five consecutive positive integers cannot be
11. a1, a2, ... , a2n are distinct integers. Find all integers x which satisfy (x -
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loga+1(b + 1). (USA 2)
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14. Let ABCD be a convex quadrilateral with the line CD tangent to the circle on diameter AB. Prove that the line AB is tangent to the circle on diameter CD if and only if BC and AD are parallel.