IMO 1984 Longlist
IMO 1984 Longlist — 48 problems.
IMO 1984 Longlist
48 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUS1 | AUS | The fraction |
| AUS2 | AUS | Given a regular convex 2m-sided polygon P, show that there is |
| AUS3 | AUS | The opposite sides of the reentrant hexagon AFBDCE in- |
| BEL4 | BEL | Given a triangle ABC, three equilateral triangles AEB, BFC, |
| BEL5 | BEL | For a real number x, let [x] denote the greatest integer not |
| BEL6 | BEL | Let P, Q, R be the polynomials with real or complex coefficients |
| BUL7 | BUL | Prove that for any natural number n, the number |
| BUL8 | BUL | In the plane of a given triangle A1A2A3 determine (with proof) |
| BUL9 | BUL | The circle inscribed in the triangle A1A2A3 is tangent to |
| BUL10 | BUL | Assume that the bisecting plane of the dihedral angle at edge |
| CAN15 | CAN | Consider all the sums of the form |
| FRA18 | FRA | Let c be the inscribed circle of the triangle ABC, d a line tan- |
| FRA19 | FRA | Let ABC be an isosceles triangle with right angle at point A. |
| FRG21 | FRG | (1) Start with a white balls and b black balls. |
| FRG23 | FRG | A 2 \times 2 \times 12 box fixed in space is to be filled with twenty-four |
| GBR26 | GBR | A cylindrical container has height 6 cm and radius 4 cm. It |
| GBR27 | GBR | The function f(n) is defined on the nonnegative integers n by: |
| GBR28 | GBR | A “number triangle” (tnk) (0 \leqk \leqn) is defined by tn,0 = |
| GDR29 | GDR | Let Sn = {1, . . . , n} and let f be a function that maps every |
| GDR30 | GDR | Decide whether it is possible to color the 1984 natural numbers |
| LUX31 | LUX | Let f1(x) = x3 +a1x2 +b1x+c1 = 0 be an equation with three |
| MON34 | MON | One country has n cities and every two of them are linked by a |
| MON35 | MON | Prove that there exist distinct natural numbers m1, m2, . . . , |
| MON36 | MON | The set {1, 2, . . ., 49} is divided into three subsets. Prove that |
| MOR37 | MOR | Denote by [x] the greatest integer not exceeding x. For all |
| MOR38 | MOR | Determine all continuous functions f such that |
| MOR39 | MOR | Let ABC be an isosceles triangle, AB = AC, \angleA = 20◦. Let |
| NET41 | NET | Determine positive integers p, q, and r such that the diagonal |
| NET42 | NET | Triangle ABC is given for which BC = AC + 1 |
| POL45 | POL | Let X be an arbitrary nonempty set contained in the plane and |
| ROM46 | ROM | Let (an)n\geq1 and (bn)n\geq1 be two sequences of natural numbers |
| ROM48 | ROM | Let ABC be a triangle with interior angle bisectors AA1, |
| ROM49 | ROM | Let n > 1 and xi \inR for i = 1, . . . , n. Set Sk = xk |
| SPA51 | SPA | Two cyclists leave simultaneously a point P in a circular run- |
| SPA52 | SPA | Construct a scalene triangle such that |
| SPA53 | SPA | Find a sequence of natural numbers ai such that ai = i+4 |
| SPA54 | SPA | Let P be a convex planar polygon with equal angles. Let |
| SPA55 | SPA | Let a, b, c be natural numbers such that a+b+c = 2pq(p30+q30), |
| SWE56 | SWE | Let a, b, c be nonnegative integers such that a \leqb \leqc, 2b ̸= |
| SWE57 | SWE | Let a, b, c, d be a permutation of the numbers 1, 9, 8, 4 and let |
| SWE58 | SWE | Let (an)\infty |
| USA59 | USA | Determine the smallest positive integer m such that 529n +m\cdot |
| USA61 | USA | A fair coin is tossed repeatedly until there is a run of an odd |
| USA62 | USA | From a point P exterior to a circle K, two rays are drawn |
| USS64 | USS | For a matrix (pij) of the format m \times n with real entries, set |
| USS65 | USS | A tetrahedron is inscribed in a sphere of radius 1 such that the |
| USS67 | USS | With the medians of an acute-angled triangle another triangle is |
| USS68 | USS | In the Martian language every finite sequence of letters of |