IMO 1991 Shortlist
IMO 1991 Shortlist — 30 problems. 30 problems.
IMO 1991 Shortlist
30 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | PHI | Let ABC be any triangle and P any point in its interior. Let |
| 2 | JAP | For an acute triangle ABC, M is the midpoint of the segment |
| 3 | PRK | Let S be any point on the circumscribed circle of \trianglePQR. Then |
| 4 | FRA | Let ABC be a triangle and M an interior point in ABC. |
| 5 | SPA | In the triangle ABC, with ∡A = 60◦, a parallel IF to AC |
| 6 | USS | Prove for each triangle ABC the inequality |
| 7 | CHN | Let O be the center of the circumsphere of a tetrahedron |
| 8 | NET | Let S be a set of n points in the plane. No three points of |
| 9 | FRA | In the plane we are given a set E of 1991 points, and certain |
| 10 | USA | Suppose G is a connected graph with n edges. Prove that |
| 11 | AUS | Prove that |
| 12 | CHN | Let S = {1, 2, 3, . . ., 280}. Find the minimal natural num- |
| 13 | POL | Given any integer n \geq2, assume that the integers a1, a2, . . . , an |
| 14 | POL | Let a, b, c be integers and p an odd prime number. Prove that |
| 15 | USS | Let an be the last nonzero digit in the decimal representation |
| 16 | ROM | Let n > 6 and a1 < a2 < \cdot \cdot \cdot < ak be all natural numbers |
| 17 | HKG | Find all positive integer solutions x, y, z of the equation 3x + |
| 18 | BUL | Find the highest degree k of 1991 for which 1991k divides the |
| 19 | IRE | Let a be a rational number with 0 < a < 1 and suppose that |
| 20 | IRE | Let \alpha be the positive root of the equation x2 = 1991x + 1. For |
| 21 | HKG | Let f(x) be a monic polynomial of degree 1991 with integer |
| 22 | USA | Real constants a, b, c are such that there is exactly one square |
| 23 | IND | Let f and g be two integer-valued functions defined on the set |
| 24 | IND | An odd integer n \geq3 is said to be “nice” if there is at least one |
| 25 | USA | Suppose that n \geq2 and x1, x2, . . . , xn are real numbers between |
| 26 | CZS | Let n \geq2 be a natural number and let the real numbers |
| 27 | POL | Determine the maximum value of the sum |
| 28 | NET | Given a real number a > 1, construct an infinite and |
| 29 | FIN | We call a set S on the real line R superinvariant if for any |
| 30 | BUL | Two students A and B are playing the following game: Each |