IMO 1986 Longlist
IMO 1986 Longlist — 59 problems.
IMO 1986 Longlist
59 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUS1 | AUS | Let k be one of the integers 2, 3, 4 and let n = 2k −1. Prove |
| AUS2 | AUS | Let ABCD be a convex quadrilateral. DA and CB meet at |
| AUS3 | AUS | A line parallel to the side BC of a triangle ABC meets AB |
| BEL4 | BEL | Find the last eight digits of the binary development of 271986. |
| BEL5 | BEL | Let ABC and DEF be acute-angled triangles. Write d = EF, |
| BEL6 | BEL | In an urn there are one ball marked 1, two balls marked 2, and |
| CAN9 | CAN | In a triangle ABC, \angleBAC = 100◦, AB = AC. A point |
| CAN10 | CAN | A set of n standard dice are shaken and randomly placed in a |
| CHN12 | CHN | Let O be an interior point of a tetrahedron A1A2A3A4. Let |
| CHN13 | CHN | Let N = {1, 2, . . ., n}, n \geq3. To each pair i, j of elements of N, |
| CHN15 | CHN | Let N = B1 \cup\cdot \cdot \cdot\cupBq be a partition of the set N of all… |
| CZS16 | CZS | Given a positive integer k, find the least integer nk for which |
| CZS17 | CZS | We call a tetrahedron right-faced if each of its faces is a right- |
| FIN19 | FIN | Let f : [0, 1] \to[0, 1] satisfy f(0) = 0, f(1) = 1 and |
| FIN20 | FIN | For any angle \alpha with 0 < \alpha < 180◦, we call a closed convex |
| FRA21 | FRA | Let AB be a segment of unit length and let C, D be variable |
| FRA22 | FRA | Let (an)n\inN be the sequence of integers defined recursively by |
| FRA23 | FRA | Let I and J be the centers of the incircle and the excircle in |
| FRA24 | FRA | Two families of parallel lines are given in the plane, consisting |
| FRG27 | FRG | In an urn there are n balls numbered 1, 2, . . . , n. They are |
| FRG29 | FRG | We define a binary operation ⋆in the plane as follows: Given |
| FRG30 | FRG | Prove that a convex polyhedron all of whose faces are equilat- |
| GBR31 | GBR | Let P and Q be distinct points in the plane of a triangle ABC |
| GBR32 | GBR | Find, with proof, all solutions of the equation 1 |
| GBR34 | GBR | For each nonnegative integer n, Fn(x) is a polynomial in x of |
| GBR35 | GBR | Establish the maximum and minimum values that the sum |
| GDR37 | GDR | Prove that the set {1, 2, . . ., 1986} can be partitioned into 27 |
| GRE39 | GRE | Let S be a k-element set. |
| GRE40 | GRE | Find the maximum value that the quantity 2m + 7n can have |
| GRE41 | GRE | Let M, N, P be the midpoints of the sides BC, CA, AB of a |
| HUN42 | HUN | The integers 1, 2, . . ., n2 are placed on the fields of an n \times n |
| IRE45 | IRE | Given n real numbers a1 \leqa2 \leq\cdot \cdot \cdot \leqan, define |
| IRE46 | IRE | We wish to construct a matrix with 19 rows and 86 columns, |
| ISR48 | ISR | Let P be a convex 1986-gon in the plane. Let A, D be interior |
| ISR49 | ISR | Let C1, C2 be circles of radius 1/2 tangent to each other and |
| LUX50 | LUX | Let D be the point on the side BC of the triangle ABC such |
| MON51 | MON | Let a, b, c, d be the lengths of the sides of a quadrilateral |
| MON52 | MON | Solve the system of equations |
| MON53 | MON | For given positive integers r, v, n let S(r, v, n) denote the num- |
| MON54 | MON | Find the least integer n with the following property: For any |
| MON55 | MON | Given an integer n \geq2, determine all n-digit numbers |
| MOR56 | MOR | Let A1A2A3A4A5A6 be a hexagon inscribed into a circle with |
| MOR57 | MOR | In a triangle ABC, the incircle touches the sides BC, CA, AB |
| NET60 | NET | Prove the inequality |
| ROM61 | ROM | Given a positive integer n, find the greatest integer p with the |
| ROM62 | ROM | Determine all pairs of positive integers (x, y) satisfying the |
| ROM63 | ROM | Let AA′, BB′, CC′ be the bisectors of the angles of a triangle |
| ROM64 | ROM | Let (an)n\inN be the sequence of integers defined recursively by |
| ROM65 | ROM | Let A1A2A3A4 be a quadrilateral inscribed in a circle C. Show |
| SWE66 | SWE | One hundred red points and one hundred blue points are |
| SWE68 | SWE | Consider the equation x4 + ax3 + bx2 + ax + 1 = 0 with real |
| TUR71 | TUR | Two straight lines perpendicular to each other meet each side |
| TUR72 | TUR | A one-person game with two possible outcomes is played as |
| TUR73 | TUR | Let (ai)i\inN be a strictly increasing sequence of positive real |
| USA75 | USA | The incenter of a triangle is the midpoint of the line seg- |
| USS77 | USS | Find all integers x, y, z that satisfy |
| USS78 | USS | If T and T1 are two triangles with angles x, y, z and x1, y1, z1, |
| USS79 | USS | Let AA1, BB1, CC1 be the altitudes in an acute-angled triangle |
| USS80 | USS | Let ABCD be a tetrahedron and O its incenter, and let the |
IMO 1986 LL CHN15
Let N = B1 \cup\cdot \cdot \cdot\cupBq be a partition of the set N of all positive