IMO 1989 Shortlist
IMO 1989 Shortlist — 32 problems. 32 problems.
IMO 1989 Shortlist
32 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | AUS | Let ABC be a triangle. The bisector of angle A meets |
| 2 | AUS | Ali Barber, the carpet merchant, has a rectangular piece of |
| 3 | AUS | Ali Barber, the carpet merchant, has a rectangular piece of |
| 4 | BUL | Prove that for every integer n > 1 the equation |
| 5 | COL | Consider the polynomial p(x) = xn+nxn−1+a2xn−2 +\cdot \cdot \cdot+an |
| 6 | CZS | For a triangle ABC, let k be its circumcircle with radius r. The |
| 7 | FIN | Show that any two points lying inside a regular n-gon E can |
| 8 | FRA | Let R be a rectangle that is the union of a finite number of |
| 9 | FRA | For all integers n, n \geq0, there exist uniquely determined |
| 10 | GRE | Let g : C \toC, w \inC, a \inC, w3 = 1 (w ̸= 1). Show that |
| 11 | HUN | Define sequence an by |
| 12 | HUN | At n distinct points of a circular race course there are n cars |
| 13 | ICE | The quadrilateral ABCD has the following properties: |
| 14 | IND | A bicentric quadrilateral is one that is both inscribable in |
| 15 | IRE | Let a, b, c, d, m, n be positive integers such that a2+b2+c2+d2 = |
| 16 | ISR | The set {a0, a1, . . . , an} of real numbers satisfies the following |
| 17 | MON | Given seven points in the plane, some of them are connected |
| 18 | MON | Given a convex polygon A1A2 . . . An with area S, and a point |
| 19 | MON | A positive integer is written in each square of an m\timesn board. |
| 20 | NET | Given a set S in the plane containing n points and satis- |
| 21 | NET | Prove that the intersection of a plane and a regular tetrahedron |
| 22 | PHI | Prove that the set {1, 2, . . ., 1989} can be expressed as the |
| 23 | POL | We consider permutations (x1, . . . , x2n) of the set {1, . . . , |
| 24 | POL | For points A1, . . . , A5 on the sphere of radius 1, what is the |
| 25 | KOR | Let a, b be integers that are not perfect squares. Prove that if |
| 26 | KOR | Let n be a positive integer and let a, b be given real numbers. |
| 27 | ROM | Let m be a positive odd integer, m \geq2. Find the smallest |
| 28 | ROM | Consider in a plane \Pi the points O, A1, A2, A3, A4 such that |
| 29 | ROM | A flock of 155 birds sit down on a circle C. Two birds Pi, Pj are |
| 30 | SWE | For which positive integers n does there exist a positive |
| 31 | SWE | Let a1 \geqa2 \geqa3 be given positive integers and let N(a1, a2, a3) |
| 32 | USA | The vertex A of the acute triangle ABC is equidistant from |