IMO 1990 Shortlist
IMO 1990 Shortlist — 28 problems. 28 problems.
IMO 1990 Shortlist
28 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | AUS | The integer 9 can be written as a sum of two consecutive |
| 2 | CAN | Given n countries with three representatives each, m commit- |
| 3 | CZS | On a circle, 2n −1 (n \geq3) different points are given. Find |
| 4 | CZS | Assume that the set of all positive integers is decomposed into |
| 5 | FRA | Given \triangleABC with no side equal to another side, let G, K, |
| 6 | FRG | Two players A and B play a game in which they choose |
| 7 | GRE | Let f(0) = f(1) = 0 and |
| 8 | HUN | For a given positive integer k denote the square of the sum of |
| 9 | HUN | The incenter of the triangle ABC is K. The midpoint of AB |
| 10 | ICE | A plane cuts a right circular cone into two parts. The plane is |
| 11 | — | (IND 3′)IMO1 Given a circle with two chords AB, CD that meet at E, let |
| 12 | IRE | Let ABC be a triangle and L the line through C parallel to |
| 13 | IRE | An eccentric mathematician has a ladder with n rungs that he |
| 14 | JAP | In the coordinate plane a rectangle with vertices (0, 0), (m, 0), |
| 15 | MEX | Determine for which positive integers k the set |
| 16 | NET | Is there a 1990-gon with the following properties (i) and |
| 17 | NET | Unit cubes are made into beads by drilling a hole through |
| 18 | NOR | Let a, b be natural numbers with 1 \leqa \leqb, and M = |
| 19 | POL | Let P be a point inside a regular tetrahedron T of unit volume. |
| 20 | POL | Prove that every integer k greater than 1 has a multiple that is |
| 21 | — | (ROM 1′) Let n be a composite natural number and p a proper divisor |
| 22 | ROM | Ten localities are served by two international airlines such |
| 23 | ROM | Find all positive integers n having the property that 2n+1 |
| 24 | THA | Let a, b, c, d be nonnegative real numbers such that ab + bc + |
| 25 | TUR | Let Q+ be the set of positive rational numbers. Construct |
| 26 | USA | Let P be a cubic polynomial with rational coefficients, and let |
| 27 | USS | Find all natural numbers n for which every natural number |
| 28 | USS | Prove that on the coordinate plane it is impossible to draw a |