IMO 1984 Shortlist
IMO 1984 Shortlist — 20 problems. 20 problems.
IMO 1984 Shortlist
20 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | FRA | Find all solutions of the following system of n equations in n |
| 2 | CAN | Prove: |
| 3 | USS | Find all positive integers n such that |
| 4 | MON | Let d be the sum of the lengths of all diagonals of a convex |
| 5 | FRG | Let x, y, z be nonnegative real numbers with x+y +z = 1. |
| 6 | CAN | Let c be a positive integer. The sequence {fn} is defined as |
| 7 | FRG | (a) Decide whether the fields of the 8 \times 8 chessboard can be numbered |
| 8 | ROM | In a plane two different points O and A are given. For |
| 9 | POL | Let a, b, c be positive numbers with \sqrta+ |
| 10 | GBR | Prove that the product of five consecutive positive integers |
| 11 | CAN | Let n be a natural number and a1, a2, . . . , a2n mutually distinct |
| 12 | NET | Find two positive integers a, b such that none of the num- |
| 13 | BUL | Prove that the volume of a tetrahedron inscribed in a right |
| 14 | ROM | Let ABCD be a convex quadrilateral for which the circle |
| 15 | LUX | Angles of a given triangle ABC are all smaller than 120◦. |
| 16 | POL | Let a, b, c, d be odd positive integers such that a < b < c < |
| 17 | FRG | In a permutation (x1, x2, . . . , xn) of the set 1, 2, . . . , n we call |
| 18 | USA | Inside triangle ABC there are three circles k1, k2, k3 each of |
| 19 | CAN | The triangular array (an,k) of numbers is given by an,1 = 1/n, |
| 20 | USA | Determine all pairs (a, b) of positive real numbers with a ̸= 1 |