IMO 1987 Shortlist
IMO 1987 Shortlist — 23 problems. 23 problems.
IMO 1987 Shortlist
23 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | AUS | Let f be a function that satisfies the following conditions: |
| 2 | USA | At a party attended by n married couples, each person talks |
| 3 | FIN | Does there exist a second-degree polynomial p(x, y) in two |
| 4 | FRA | Let ABCDEFGH be a parallelepiped with AE\parallelBF\parallelCG\parallelDH. |
| 5 | GBR | Find, with proof, the point P in the interior of an acute-angled |
| 6 | GRE | Show that if a, b, c are the lengths of the sides of a triangle |
| 7 | NET | Given five real numbers u0, u1, u2, u3, u4, prove that it is always |
| 8 | HUN | (a) Let (m, k) = 1. Prove that there exist integers a1, a2, . . . , am |
| 9 | HUN | Does there exist a set M in usual Euclidean space such that |
| 10 | ICE | Let S1 and S2 be two spheres with distinct radii that touch |
| 11 | POL | Find the number of partitions of the set {1, 2, . . ., n} into three |
| 12 | POL | Given a nonequilateral triangle ABC, the vertices listed coun- |
| 13 | GDR | Is it possible to put 1987 points in the Euclidean plane |
| 14 | FRG | How many words with n digits can be formed from the alphabet |
| 15 | FRG | Suppose x1, x2, . . . , xn are real numbers with x2 |
| 16 | FRG | Let S be a set of n elements. We denote the number of all |
| 17 | ROM | Prove that there exists a four-coloring of the set M = |
| 18 | ROM | For any integer r \geq1, determine the smallest integer h(r) \geq1 |
| 19 | USS | Let \alpha, \beta, \gamma be positive real numbers such that \alpha + \beta +… |
| 20 | USS | Let f(x) = x2 + x + p, p \inN. Prove that if the numbers |
| 21 | USS | The prolongation of the bisector AL (L \inBC) in the acute- |
| 22 | VIE | Does there exist a function f : N \toN, such that f(f(n)) = |
| 23 | YUG | Prove that for every natural number k (k \geq2) there exists an |