IMO 1993 Shortlist
IMO 1993 Shortlist — 26 problems. 26 problems.
IMO 1993 Shortlist
26 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | BRA | Show that there exists a finite set A \subsetR2 such that for |
| 2 | CAN | Let triangle ABC be such that its circumradius R is equal to |
| 3 | SPA | Consider the triangle ABC, its circumcircle k with center O |
| 4 | SPA | In the triangle ABC, let D, E be points on the side BC such |
| 5 | FIN | On an infinite chessboard, a solitaire game is played as |
| 6 | GER | Let N = {1, 2, 3, . . .}. Determine whether there exists a |
| 7 | GEO | Let a, b, c be given integers a > 0, ac −b2 = P = P1 \cdot \cdot \cdot Pm |
| 8 | IND | Define a sequence ⟨f(n)⟩\infty |
| 9 | IND | (a) Show that the set Q+ of all positive rational numbers can be par- |
| 10 | IND | A natural number n is said to have the property P if whenever |
| 11 | IRE | Let n > 1 be an integer and let f(x) = xn + 5xn−1 + 3. |
| 12 | IRE | Let n, k be positive integers with k \leqn and let S be a set |
| 13 | IRE | Let S be the set of all pairs (m, n) of relatively prime positive |
| 14 | ISR | The vertices D, E, F of an equilateral triangle lie on the sides |
| 15 | MCD | For three points A, B, C in the plane we define m(ABC) |
| 16 | MCD | Let n \inN, n \geq2, and A0 = (a01, a02, . . . , a0n) be any n-tuple |
| 17 | NET | Let n be an integer greater than 1. In a circular arrange- |
| 18 | POL | Let Sn be the number of sequences (a1, a2, . . . , an), where ai \in |
| 19 | ROM | Let a, b, n be positive integers, b > 1 and bn −1 | a. Show |
| 20 | ROM | Let c1, . . . , cn \inR (n \geq2) such that 0 \leqn |
| 21 | GBR | A circle S is said to cut a circle \Sigma diametrally if their common |
| 22 | GBR | A, B, C, D are four points in the plane, with C, D on the |
| 23 | GBR | A finite set of (distinct) positive integers is called a “DS-set” |
| 24 | USA | Prove that |
| 25 | VIE | Solve the following system of equations, in which a is a given |
| 26 | VIE | Let a, b, c, d be four nonnegative numbers satisfying a+b+c+d = |