IMO 1997 Shortlist
IMO 1997 Shortlist — 26 problems. 26 problems.
IMO 1997 Shortlist
26 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | BLR | An infinite square grid is colored in the chessboard pattern. |
| 2 | CAN | Let R1, R2, . . . be the family of finite sequences of positive inte- |
| 3 | GER | For each finite set U of nonzero vectors in the plane we define |
| 4 | IRN | An n \times n matrix with entries from {1, 2, . . ., 2n −1} is called |
| 5 | ROM | Let ABCD be a regular tetrahedron and M, N distinct points |
| 6 | IRE | (a) Let n be a positive integer. Prove that there exist distinct |
| 7 | RUS | Let ABCDEF be a convex hexagon such that AB = BC, CD = |
| 8 | GBR | Four different points A, B, C, D are chosen on a circle \Gamma such |
| 9 | USA | Let A1A2A3 be a nonisosceles triangle with incenter I. Let Ci, |
| 10 | CZE | Find all positive integers k for which the following statement is |
| 11 | NET | Let P(x) be a polynomial with real coefficients such that P(x) > |
| 12 | ITA | Let p be a prime number and let f(x) be a polynomial of degree |
| 13 | IND | In town A, there are n girls and n boys, and each girl knows each |
| 14 | IND | Let b, m, n be positive integers such that b > 1 and m ̸= n. Prove |
| 15 | RUS | An infinite arithmetic progression whose terms are positive in- |
| 16 | BLR | In an acute-angled triangle ABC, let AD, BE be altitudes and |
| 17 | CZE | Find all pairs of integers x, y \geq1 satisfying the equation |
| 18 | GBR | The altitudes through the vertices A, B, C of an acute-angled |
| 19 | IRE | Let a1 \geq\cdot \cdot \cdot \geqan \geqan+1 = 0 be a sequence of real numbers. |
| 20 | IRE | Let D be an internal point on the side BC of a triangle ABC. |
| 21 | RUS | Let x1, x2, . . . , xn be real numbers satisfying the conditions |
| 22 | UKR | (a) Do there exist functions f : R \toR and g : R \toR such that |
| 23 | GBR | Let ABCD be a convex quadrilateral and O the intersection of |
| 24 | LIT | For a positive integer n, let f(n) denote the number of ways to |
| 25 | POL | The bisectors of angles A, B, C of a triangle ABC meet its cir- |
| 26 | ITA | For every integer n \geq2 determine the minimum value that the |