IMO 1998 Shortlist
IMO 1998 Shortlist — 28 problems. 28 problems.
IMO 1998 Shortlist
28 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | LUX | A convex quadrilateral ABCD has perpendicular diagonals. |
| 2 | POL | Let ABCD be a cyclic quadrilateral. Let E and F be variable |
| 3 | UKR | Let I be the incenter of triangle ABC. Let K, L, and M |
| 4 | ARM | Let M and N be points inside triangle ABC such that |
| 5 | FRA | Let ABC be a triangle, H its orthocenter, O its circumcenter, |
| 6 | POL | Let ABCDEF be a convex hexagon such that \angleB +\angleD +\angleF = |
| 7 | GBR | Let ABC be a triangle such that \angleACB = 2\angleABC. Let D be |
| 8 | IND | Let ABC be a triangle such that \angleA = 90◦and \angleB < \angleC. The |
| 9 | MON | Let a1, a2, . . . , an be positive real numbers such that a1 + a2 + |
| 10 | AUS | Let r1, r2, . . . , rn be real numbers greater than or equal to 1. |
| 11 | RUS | Let x, y, and z be positive real numbers such that xyz = 1. Prove |
| 12 | POL | Let n \geqk \geq0 be integers. The numbers c(n, k) are defined as |
| 13 | BUL | Determine the least possible value of f(1998), where f is a |
| 14 | GBR | Determine all pairs (x, y) of positive integers such that x2y+ |
| 15 | AUS | Determine all pairs (a, b) of real numbers such that a\lfloorbn\rfloor=… |
| 16 | UKR | Determine the smallest integer n \geq4 for which one can choose |
| 17 | GBR | A sequence of integers a1, a2, a3, . . . is defined as follows: a1 = 1, |
| 18 | BUL | Determine all positive integers n for which there exists an integer |
| 19 | BLR | For any positive integer n, let \tau(n) denote the number of its |
| 20 | ARG | Prove that for each positive integer n, there exists a positive |
| 21 | CAN | Let a0, a1, a2, . . . be an increasing sequence of nonnegative inte- |
| 22 | UKR | A rectangular array of numbers is given. In each row and each |
| 23 | BLR | Let n be an integer greater than 2. A positive integer is said to be |
| 24 | SWE | Cards numbered 1 to 9 are arranged at random in a row. In a |
| 25 | NZL | Let U = {1, 2, . . ., n}, where n \geq3. A subset S of U is said to be |
| 26 | IND | In a contest, there are m candidates and n judges, where |
| 27 | BLR | Ten points such that no three of them lie on a line are marked in |
| 28 | IRN | A solitaire game is played on an m \times n rectangular board, using |