IMO 1982 Longlist
IMO 1982 Longlist — 37 problems.
IMO 1982 Longlist
37 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUS1 | AUS | It is well known that the binomial coefficients |
| AUS2 | AUS | Given a finite number of angular regions A1, . . . , Ak in a plane, |
| AUS3 | AUS | Given n points X1, X2, . . . , Xn in the interval 0 \leqXi \leq1, |
| BEL5 | BEL | Among all triangles with a given perimeter, find the one with |
| BEL6 | BEL | On the three distinct lines a, b, and c three points A, B, and |
| BEL7 | BEL | Find all solutions (x, y) \inZ2 of the equation |
| BRA9 | BRA | Let n be a natural number, n \geq2, and let \varphi be Euler’s function; |
| BRA10 | BRA | Let r1, . . . , rn be the radii of n spheres. Call S1, S2, . . . , Sn the |
| BRA11 | BRA | A rectangular pool table has a hole at each of three of its |
| BRA12 | BRA | Let there be 3399 numbers arbitrarily chosen among the first |
| BUL13 | BUL | A regular n-gonal truncated pyramid is circumscribed around |
| CAN15 | CAN | Show that the set S of natural numbers n for which 3/n |
| CAN18 | CAN | You are given an algebraic system admitting addition and |
| CZS20 | CZS | Consider a cube C and two planes \sigma, \tau, which divide Euclidean |
| CZS21 | CZS | All edges and all diagonals of regular hexagon A1A2A3A4A5A6 |
| FIN23 | FIN | Determine the sum of all positive integers whose digits (in base |
| FIN24 | FIN | Prove that if a person a has infinitely many descendants (chil- |
| FRA26 | FRA | Let (an)n\geq0 and (bn)n\geq0 be two sequences of natural numbers. |
| FRA28 | FRA | Let (u1, . . . , un) be an ordered ntuple. For each k, 1 \leqk \leqn, |
| FRA29 | FRA | Let f : R \toR be a continuous function. Suppose that the |
| GBR33 | GBR | A sequence (un) of integers is defined for n \geq0 by u0 = 0, |
| GDR34 | GDR | Let M be the set of all functions f with the following proper- |
| GDR35 | GDR | If the inradius of a triangle is half of its circumradius, prove |
| POL38 | POL | Numbers un,k (1 \leqk \leqn) are defined as follows: |
| POL39 | POL | Let S be the unit circle with center O and let P1, P2, . . . , Pn |
| POL40 | POL | We consider a game on an infinite chessboard similar to that of |
| POL42 | POL | Let F be the family of all k-element subsets of the set |
| TUN43 | TUN | (a) What is the maximal number of acute angles in a convex |
| TUN44 | TUN | Let A and B be positions of two ships M and N, respectively, |
| USA46 | USA | Prove that if a diagonal is drawn in a quadrilateral inscribed |
| USA47 | USA | Evaluate sec′′ \pi |
| USA48 | USA | Given a finite sequence of complex numbers c1, c2, . . . , cn, show |
| USA49 | USA | Simplify |
| USS50 | USS | Let O be the midpoint of the axis of a right circular cylinder. |
| USS51 | USS | Let n numbers x1, x2, . . . , xn be chosen in such a way that |
| USS52 | USS | We are given 2n natural numbers |
| VIE56 | VIE | Let f(x) = ax2 + bx + c and g(x) = cx2 + bx + a. If |f(0)| \leq1, |