### 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, ... , an, b1, b2, ... , bn}. There are subsets C1, C2, ... , Ck, such that (1) for no i, j do both ai and bi both belong to Cj, (2) for any pair of distinct elements of S, not of the form ai, bi, there is just one Cj 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 x11/2 + x21/2 + ... + xn1/2 ≤ c (x1 + x2 + ... + xn)1/2 for all n and all real sequences x1, x2, x3, ... which satisfy x1 + x2 + ... + xn ≤ xn+1. (United Kingdom 6) 6.  Show that an/(b+c) + bn/(c+a) + cn/(a+b) ≥ (2/3)n-2 sn-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 ui - vi is integral and ∑i 1, there is an irrational number r such that [rm] = -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.

Shortlist home