IMO 1985 Longlist
IMO 1985 Longlist — 75 problems.
IMO 1985 Longlist
75 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUS2 | AUS | We are given a triangle ABC and three rectangles R1, R2, R3 |
| AUS3 | AUS | A function f has the following property: If k > 1, j > 1, |
| BEL4 | BEL | Let x, y, and z be real numbers satisfying x + y + z = xyz. |
| BEL6 | BEL | On a one-way street, an unending sequence of cars of width a, |
| BRA7 | BRA | A convex quadrilateral is inscribed in a circle of radius 1. Prove |
| BRA8 | BRA | Let K be a convex set in the xy-plane, symmetric with respect |
| BUL11 | BUL | Let a and b be integers and n a positive integer. Prove that |
| CAN12 | CAN | Find the maximum value of |
| CAN13 | CAN | Find the average of the quantity |
| CAN14 | CAN | Let k be a positive integer. Define u0 = 0, u1 = 1, and |
| CAN15 | CAN | Superchess is played on on a 12 \times 12 board, and it uses su- |
| CUB17 | CUB | Set |
| CYP18 | CYP | The circles (R, r) and (P, \rho), where r > \rho, touch externally |
| CYP19 | CYP | Solve the system of simultaneous equations |
| CZS20 | CZS | Let T be the set of all lattice points (i.e., all points with |
| CZS21 | CZS | Let A be a set of positive integers such that for any two elements |
| CZS23 | CZS | Let N = {1, 2, 3, . . .}. For real x, y, set S(x, y) = {s | s = |
| FRA24 | FRA | Let d \geq1 be an integer that is not the square of an integer. |
| FRA25 | FRA | Find eight positive integers n1, n2, . . . , n8 with the follow- |
| FRA27 | FRA | Let O be a point on the oriented Euclidean plane and (i, j) |
| FRG28 | FRG | Let M be the set of the lengths of an octahedron whose sides |
| FRG29 | FRG | Call a four-digit number (xyzt)B in the number system with |
| GBR30 | GBR | A plane rectangular grid is given and a “rational point” is |
| GBR31 | GBR | Let E1, E2, and E3 be three mutually intersecting ellipses, all |
| GBR32 | GBR | A collection of 2n letters contains 2 each of n different letters. |
| GDR35 | GDR | We call a coloring f of the elements in the set M = {(x, y) | |
| GDR36 | GDR | Determine whether there exist 100 distinct lines in the plane |
| GDR37 | GDR | Prove that a triangle with angles \alpha, \beta, \gamma, circumradius R, and |
| IRE39 | IRE | Given a triangle ABC and external points X, Y , and Z such |
| IRE40 | IRE | Each of the numbers x1, x2, . . . , xn equals 1 or −1 and |
| ISR42 | ISR | Prove that the product of two sides of a triangle is always |
| ISR43 | ISR | Suppose that 1985 points are given inside a unit cube. Show |
| ITA45 | ITA | Two persons, X and Y , play with a die. X wins a game if the |
| ITA46 | ITA | Let C be the curve determined by the equation y = x3 in the |
| ITA47 | ITA | Let F be the correspondence associating with every point P = |
| ITA48 | ITA | In a given country, all inhabitants are knights or knaves. A |
| MON50 | MON | From each of the vertices of a regular n-gon a car starts to |
| MON51 | MON | Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers. |
| MON52 | MON | In the triangle ABC, let B1 be on AC, E on AB, G on BC, |
| MON53 | MON | For each P inside the triangle ABC, let A(P), B(P), and |
| MOR54 | MOR | Set Sn = n |
| MOR55 | MOR | The points A, B, C are in this order on line D, and AB = 4BC. |
| MOR56 | MOR | Let ABCD be a rhombus with angle \angleA = 60◦. Let E be a |
| NET57 | NET | The solid S is defined as the intersection of the six spheres with |
| NET58 | NET | Prove that there are infinitely many pairs (k, N) of positive |
| NOR60 | NOR | The sequence (sn), where sn = n |
| NOR61 | NOR | Consider the set A = {0, 1, 2, . . ., 9} and let (B1, B2, . . . , Bk) |
| NOR62 | NOR | A “large” circular disk is attached to a vertical wall. It rotates |
| POL64 | POL | Let p be a prime. For which k can the set {1, 2, . . ., k} be |
| POL65 | POL | Define the functions f, F : N \toN, by |
| ROM67 | ROM | Let k \geq2 and n1, n2, . . . , nk \geq1 natural numbers having the |
| ROM68 | ROM | Show that the sequence {an}n\geq1 defined by an = [n |
| ROM69 | ROM | Let A and B be two finite disjoint sets of points in the plane |
| ROM70 | ROM | Let C be a class of functions f : N \toN that contains the |
| ROM71 | ROM | For every integer r > 1 find the smallest integer h(r) > 1 |
| SPA72 | SPA | Construct a triangle ABC given the side AB and the distance |
| SPA73 | SPA | Let A1A2, B1B2, C1C2 be three equal segments on the three |
| SPA74 | SPA | Find the triples of positive integers x, y, z satisfying |
| SPA75 | SPA | Let ABCD be a rectangle, AB = a, BC = b. Consider the |
| SWE76 | SWE | Are there integers m and n such that |
| SWE77 | SWE | Two equilateral triangles are inscribed in a circle with radius |
| SWE79 | SWE | Let a, b, and c be real numbers such that |
| TUR80 | TUR | Let E = {1, 2, . . ., 16} and let M be the collection of all |
| TUR81 | TUR | Given the side a and the corresponding altitude ha of a triangle |
| TUR82 | TUR | Find all cubic polynomials x3 + ax2 + bx + c admitting the |
| TUR83 | TUR | Let \Gammai, i = 0, 1, 2, . . ., be a circle of radius ri inscribed in an |
| USA85 | USA | Let CD be a diameter of circle K. Let AB be a chord that is |
| USA86 | USA | Let l denote the length of the smallest diagonal of all rectangles |
| USA88 | USA | Determine the range of w(w + x)(w + y)(w + z), where x, y, |
| USA89 | USA | Given that n elements a1, a2, . . . , an are organized into n pairs |
| USS90 | USS | Decompose the number 51985−1 into a product of three integers, |
| USS91 | USS | Thirty-four countries participated in a jury session of the IMO, |
| USS93 | USS | The sphere inscribed in tetrahedron ABCD touches the sides |
| VIE96 | VIE | Determine all functions f : R \toR satisfying the following two |
| VIE97 | VIE | In a plane a circle with radius R and center w and a line \Lambda |