28th IMO 1987 shortlist

28th IMO 1987 shortlisted problems


1. f is a real-valued function on the reals such that: (1) if x ≥ y and f(y) - y ≥ v ≥ f(x) - x, then f(z) = v + z for some z between x and y; (2) for some k, f(k) = 0 and if f(h) = 0, then h <= k; (3) f(0) = 1; (4) f(1987) ≤ 1988; (5) f(x) f(y) = f(x f(y) + y f(x) - xy) for all x, y. Find f(1987). (Australia 6)
2. S = {a₁, a₂, ... , a_n, b₁, b₂, ... , b_n}. There are subsets C₁, C₂, ... , C_k, such that (1) for no i, j do both a_i and b_i both belong to C_j, (2) for any pair of distinct elements of S, not of the form a_i, b_i, there is just one C_j containing both elements. Show that if n > 3, then k ≥ 2n. (USA 3)
3. Does there exist a polynomial p(x, y) of degree 2 such that, for each non-negative integer n, we have n = p(a, b) for just one pair (a, b) of non-negative integers? (Finland 3)
4. ABCDA'B'C'D' is any parallelepiped (with ABCD, A'B'C'D' faces and AA', BB', CC', DD' edges). Show that AC + AB'+ AD' <= AB + AD + AA' + AC' (the sum of the thre short diagonals from A is less than the sum of the three edges from A plus the long diagonal from A). (France 5)
5. Find the smallest real c such that x₁^1/2 + x₂^1/2 + ... + x_n^1/2 ≤ c (x₁ + x₂ + ... + x_n)^1/2 for all n and all real sequences x₁, x₂, x₃, ... which satisfy x₁ + x₂ + ... + x_n ≤ x_n+1. (United Kingdom 6)
6. Show that aⁿ/(b+c) + bⁿ/(c+a) + cⁿ/(a+b) ≥ (2/3)^n-2 s^n-1 for all n ≥ 1, where a, b, c are the sides of a triangle and s = (a + b + c)/2. (Greece 4)
7. Given any 5 real numbers u₀, u₁, u₂, u₃, u₄, show that we can always find 5 real numbers v₀, v₁, v₂, v₃, v₄ such that each u_i - v_i is integral and ∑_i<j(v_i - v_j)² < 4. (Netherlands 1)
8. Does there exist a subset M of Euclidean space such that any plane meets M in a finite non-empty set? (Hungary 1)
9. Show that for any relatively prime positive integers m, n we can find integers a₁, a₂, ... , a_m and b₁, b₂, ... , b_n such that each product a_ib_j gives a different residue mod mn. (Hungary 2)
10. Two spheres S and S' touch externally and lie inside a cone C. Each sphere touches the cone in a full circle. n solid spheres are arranged in the cone in a ring so that each touches S and S' externally, touches the cone, and touches its two neighbouring solid spheres. What are the possible values of n? (Iceland 3)
11. Find the number of ways of partitioning {1, 2, 3, ... , n} into three (possibly empty) subsets A, B, C such that (1) for each subset, if the elements are written in ascending order, then they alternate in parity, and (2) if all three subsets are non-empty, then just one of them has its smallest element even. (Poland 1)
12. ABC is a non-equilateral triangle. Find the locus of the centroid of all equilateral triangles A'B'C' such that A, B', C' are collinear, A', B, C' are collinear, A', B', C are collinear, and both ABC and A'B'C' have their vertices anti-clockwise. (Poland 5)
14. How many n-digit words can be formed from the alphabet {0, 1, 2, 3, 4} if neighboring digits must differ by exactly 1? (German Federal Republic 1)
17. Show that we can color the elements of the set {1, 2, ... , 1987} with 4 colors so that any arithmetic progression with 10 terms in the set is not monochromatic. (Romania 1)
18. For any positive integer r find the smallest positive integer h(r) such that for any partition of {1, 2, ... , h(r) } into r parts, there are integers a ≥ 0 and 1 ≤ x ≤ y such that a + x, a + y and a + x + y all belong to the same part. (Romania 4)
19. Given angles A, B, C such that A + B + C < 180^o, show that there is a triangle with sides sin A, sin B, sin C and that its area is less than (sin 2A + sin 2B + sin 2C)/8. (USSR 2)
23. Show that for any integer k > 1, there is an irrational number r such that [r^m] = -1 mod k for every natural number m. (Yugoslavia 2)
Note: 13 (German Democratic Republic 2), 15 (German Federal Republic 2), 16 (German Federal Republic 3), 20 (USSR 2), 21 (USSR 4), and 22 (Vietnam 4) do not appear here, because they were used in the Olympiad.