IMO 1989 Longlist
IMO 1989 Longlist — 77 problems.
IMO 1989 Longlist
77 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUS1 | AUS | In the set Sn = {1, 2, . . ., n} a new multiplication a∗b is defined |
| BUL5 | BUL | The sequences a0, a1, . . . and b0, b1, . . . are defined by the equal- |
| BUL6 | BUL | The circles c1 and c2 are tangent at the point A. A straight |
| COL9 | COL | Let m be a positive integer and define f(m) to be the number |
| CUB10 | CUB | Given the equation |
| CUB11 | CUB | Given the equation |
| CUB12 | CUB | Let P(x) be a polynomial such that the following inequalities |
| CUB13 | CUB | Let n be a natural number not greater than 44. Prove that for |
| CZS15 | CZS | A sequence a1, a2, a3, . . . is defined recursively by a1 = 1 and |
| FIN17 | FIN | Let a, 0 < a < 1, be a real number and f a continuous function |
| FIN18 | FIN | There are some boys and girls sitting in an n \times n quadratic |
| FRA19 | FRA | Let a1, . . . , an be distinct positive integers that do not contain |
| FRA22 | FRA | Let ABC be an equilateral triangle with side length equal to a |
| GBR25 | GBR | Let ABC be a triangle. Prove that there is a unique point U |
| GBR26 | GBR | Let a, b, c, d be positive integers such that ab = cd and a + b = |
| GBR27 | GBR | Integers cm,n (m \geq0, n \geq0) are defined by cm,0 = 1 for all |
| GBR28 | GBR | Let b1, b2, . . . , b1989 be positive real numbers such that the |
| GRE29 | GRE | Let L denote the set of all lattice points of the plane (points |
| GRE30 | GRE | In a triangle ABC for which 6(a + b + c)r2 = abc, we consider |
| HKG32 | HKG | Let ABC be an equilateral triangle. Let D, E, F, M, N, and |
| HKG33 | HKG | Let n be a positive integer. Show that ( |
| HKG34 | HKG | Given an acute triangle find a point inside the triangle such |
| HKG35 | HKG | Find all square numbers S1 and S2 such that S1 −S2 = 1989. |
| HKG36 | HKG | Prove the identity |
| HUN38 | HUN | Connecting the vertices of a regular n-gon we obtain a closed |
| ICE40 | ICE | A sequence of real numbers x0, x1, x2, . . . is defined as follows: |
| ICE41 | ICE | Alice has two urns. Each urn contains four balls and on each |
| INA43 | INA | Let f(x) = a sin2 x+b sin x+c, where a, b, and c are real num- |
| INA44 | INA | Let A and B be fixed distinct points on the X axis, none of |
| INA45 | INA | The expressions a + b + c, ab + ac + bc, and abc are called the |
| INA46 | INA | Given two distinct numbers b1 and b2, their product can be |
| INA47 | INA | Let log2 |
| INA48 | INA | Let S be the point of intersection of the two lines l1 : 7x−5y + |
| IND49 | IND | Let A, B denote two distinct fixed points in space. Let X, P |
| IND51 | IND | Let t(n), for n = 3, 4, 5, . . ., represent the number of distinct, |
| IRE53 | IRE | Let f(x) = (x −a1)(x −a2) \cdot \cdot \cdot (x −an) −2, where n \geq3 |
| IRE54 | IRE | Let f be a function from the real numbers to the real numbers |
| IRE55 | IRE | Let [x] denote the greatest integer less than or equal to x. Let \alpha |
| IRE56 | IRE | Let n = 2k −1, where k \geq6 is an integer. Let T be the set |
| ISR58 | ISR | Let P1(x), P2(x), . . . , Pn(x) be polynomials with real coefficients. |
| ISR59 | ISR | Let v1, v2, . . . , v1989 be a set of coplanar vectors with |vr| \leq1 |
| KOR60 | KOR | A real-valued function f on Q satisfies the following conditions |
| KOR61 | KOR | Let A be a set of positive integers such that no positive integer |
| KOR64 | KOR | Let a regular (2n + 1)-gon be inscribed in a circle of radius r. |
| LUX65 | LUX | A regular n-gon A1A2A3 . . . Ak . . . An inscribed in a circle of |
| MON67 | MON | A family of sets A1, A2, . . . , An has the following properties: |
| MON68 | MON | If 0 < k \leq1 and ai are positive real numbers, i = 1, 2, . . . , n, |
| MON70 | MON | Three mutually nonparallel lines li (i = 1, 2, 3) are given |
| MOR72 | MOR | Let ABCD be a quadrilateral inscribed in a circle with diam- |
| PHI76 | PHI | Let k and s be positive integers. For sets of real numbers |
| POL77 | POL | Given that |
| POL79 | POL | To each pair (x, y) of distinct elements of a finite set X a number |
| POL80 | POL | We are given a finite collection of segments in the plane, of |
| POR82 | POR | Solve in the set of real numbers the equation 3x3 −[x] = 3, |
| POR83 | POR | Poldavia is a strange kingdom. Its currency unit is the bourbaki |
| POR84 | POR | Let a, b, c, r, and s be real numbers. Show that if r is a root of |
| POR85 | POR | Let P(x) be a polynomial with integer coefficients such that |
| POR86 | POR | Given two natural numbers w and n, the tower of n w’s is the |
| POR87 | POR | A balance has a left pan, a right pan, and a pointer that moves |
| ROM90 | ROM | Prove that the sequence (an)n\geq0, an = [n |
| ROM92 | ROM | Find the set of all a \inR for which there is no infinite sequence |
| ROM93 | ROM | For \Phi : N \toZ let us define M\Phi = {f : N \toZ; f(x) > |
| SWE94 | SWE | Prove that a < b implies that a3 −3a \leqb3 −3b + 4. When |
| THA97 | THA | Let n be a positive integer, X = {1, 2, . . ., n}, and k a positive |
| THA98 | THA | Let f : N \toN be such that |
| THA99 | THA | An arithmetic function is a real-valued function whose do- |
| THA100 | THA | Let A be an n\timesn matrix whose elements are nonnegative real |
| TUR101 | TUR | Let ABC be an equilateral triangle and \Gamma the semicircle |
| TUR102 | TUR | If in a convex quadrilateral ABCD, E and F are the midpoints |
| USA103 | USA | An accurate 12-hour analog clock has an hour hand, a minute |
| USA104 | USA | For each nonzero complex number z, let arg z be the unique |
| USA106 | USA | Let n > 1 be a fixed integer. Define functions f0(x) = 0, |
| VIE107 | VIE | Let E be the set of all triangles whose only points with integer |
| VIE108 | VIE | For every sequence (x1, x2, . . . , xn) of the numbers {1, 2, . . ., n} |
| VIE109 | VIE | Let Ax, By be two noncoplanar rays with AB as a common per- |
| VIE110 | VIE | Do there exist two sequences of real numbers {ai}, {bi}, i \in |
| VIE111 | VIE | Find the greatest number c such that for all natural numbers |