IMO 1983 Shortlist
IMO 1983 Shortlist — 25 problems. 25 problems.
IMO 1983 Shortlist
25 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | AUS | The localities P1, P2, . . . , P1983 are served by ten international |
| 2 | BEL | Let n be a positive integer. Let \sigma(n) be the sum of the natural |
| 3 | BEL | We say that a set E of points of the Euclidian plane is |
| 4 | BEL | On the sides of the triangle ABC, three similar isosceles tri- |
| 5 | BRA | Consider the set of all strictly decreasing sequences of n natural |
| 6 | CAN | Suppose that {x1, x2, . . . , xn} are positive integers for which |
| 7 | CAN | Let a be a positive integer and let {an} be defined by a0 = 0 |
| 8 | SPA | In a test, 3n students participate, who are located in three |
| 9 | USA | If a, b, and c are sides of a triangle, prove that |
| 10 | FIN | Let p and q be integers. Show that there exists an interval I of |
| 11 | — | (FIN 2′) Let f : [0, 1] \toR be continuous and satisfy: |
| 12 | GBR | Find all functions f defined on the positive real numbers |
| 13 | LUX | Let E be the set of 19833 points of the space R3 all three |
| 14 | POL | Prove or disprove: From the interval [1, . . . , 30000] one |
| 15 | POL | Decide whether there exists a set M of natural numbers satis- |
| 16 | GDR | Let F(n) be the set of polynomials P(x) = a0+a1x+\cdot \cdot \cdot+anxn, |
| 17 | GDR | Let P1, P2, . . . , Pn be distinct points of the plane, n \geq2. Prove |
| 18 | FRG | Let a, b, c be positive integers satisfying (a, b) = (b, c) = |
| 19 | ROM | Let (Fn)n\geq1 be the Fibonacci sequence F1 = F2 = 1, Fn+2 = |
| 20 | ROM | Solve the system of equations |
| 21 | SWE | Find the greatest integer less than or equal to 21983 |
| 22 | SWE | Let n be a positive integer having at least two different prime |
| 23 | USS | Let K be one of the two intersection points of the circles W1 |
| 24 | USS | Let dn be the last nonzero digit of the decimal representation |
| 25 | USS | Prove that every partition of 3-dimensional space into three |