IMO 1983 Longlist
IMO 1983 Longlist — 51 problems.
IMO 1983 Longlist
51 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUS2 | AUS | Seventeen cities are served by four airlines. It is noted that |
| AUS3 | AUS | (a) Given a tetrahedron ABCD and its four altitudes (i.e., |
| BEL5 | BEL | Consider the set Q2 of points in R2, both of whose coordinates |
| BEL7 | BEL | Find all numbers x \inZ for which the number |
| BRA10 | BRA | Which of the numbers 1, 2, . . ., 1983 has the largest number of |
| BRA11 | BRA | A boy at point A wants to get water at a circular lake and |
| BRA12 | BRA | The number 0 or 1 is to be assigned to each of the n vertices |
| BUL13 | BUL | Let p be a prime number and a1, a2, . . . , a(p+1)/2 different nat- |
| BUL14 | BUL | Let l be tangent to the circle k at B. Let A be a point on k |
| CAN15 | CAN | Find all possible finite sequences {n0, n1, n2, . . . , nk} of integers |
| CAN17 | CAN | In how many ways can 1, 2, . . . , 2n be arranged in a 2 \times n |
| CAN18 | CAN | Let b \geq2 be a positive integer. |
| COL20 | COL | Let f and g be functions from the set A to the same set A. |
| COL21 | COL | Prove that there are infinitely many positive integers n for |
| CUB22 | CUB | Does there exist an infinite number of sets C consisting of 1983 |
| FIN24 | FIN | Every x, 0 \leqx \leq1, admits a unique representation x = |
| FRG25 | FRG | How many permutations a1, a2, . . . , an of {1, 2, . . ., n} are |
| FRG26 | FRG | Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) = |
| GBR28 | GBR | Show that if the sides a, b, c of a triangle satisfy the equation |
| GBR29 | GBR | Let O be a point outside a given circle. Two lines OAB, OCD |
| GBR30 | GBR | Prove the existence of a unique sequence {un} (n = 0, 1, 2 . . .) |
| GBR32 | GBR | Let a, b, c be positive real numbers and let [x] denote the |
| GDR34 | GDR | In a plane are given n points Pi (i = 1, 2, . . . , n) and two |
| ISR36 | ISR | The set X has 1983 members. There exists a family of subsets |
| ISR37 | ISR | The points A1, A2, . . . , A1983 are set on the circumference of a |
| KUW38 | KUW | Let {un} be the sequence defined by its first two terms u0, u1 |
| KUW39 | KUW | If \alpha is the real root of the equation |
| LUX40 | LUX | Four faces of tetrahedron ABCD are congruent triangles whose |
| LUX42 | LUX | Consider the square ABCD in which a segment is drawn |
| LUX43 | LUX | Given a square ABCD, let P, Q, R, and S be four variable |
| LUX44 | LUX | We are given twelve coins, one of which is a fake with a different |
| LUX45 | LUX | Let two glasses, numbered 1 and 2, contain an equal quantity |
| LUX46 | LUX | Let f be a real-valued function defined on I = (0, +\infty) and |
| NET47 | NET | In a plane, three pairwise intersecting circles C1, C2, C3 with |
| NET48 | NET | Prove that in any parallelepiped the sum of the lengths of the |
| POL49 | POL | Given positive integers k, m, n with km \leqn and nonnegative |
| ROM53 | ROM | Let a \inR and let z1, z2, . . . , zn be complex numbers of mod- |
| ROM55 | ROM | For every a \inN denote by M(a) the number of elements of |
| ROM56 | ROM | Consider the expansion |
| SPA57 | SPA | In the system of base n2 + 1 find a number N with n different |
| SPA59 | SPA | Solve the equation |
| SWE61 | SWE | Let a and b be integers. Is it possible to find integers p and q |
| SWE62 | SWE | A circle \gamma is drawn and let AB be a diameter. The point C |
| USA64 | USA | The sum of all the face angles about all of the vertices except |
| USA65 | USA | Let ABCD be a convex quadrilateral whose diagonals AC and |
| USA67 | USA | The altitude from a vertex of a given tetrahedron intersects |
| USA68 | USA | Three of the roots of the equation x4 −px3 + qx2 −rx + s = 0 |
| USS72 | USS | Prove that for all x1, x2, . . . , xn \inR the following inequality |
| VIE73 | VIE | Let ABC be a nonequilateral triangle. Prove that there exist |
| VIE74 | VIE | In a plane we are given two distinct points A, B and two lines |
| VIE75 | VIE | Find the sum of the fiftieth powers of all sides and diagonals of |