IMO 1971 Longlist
IMO 1971 Longlist — 38 problems.
IMO 1971 Longlist
38 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUT1 | AUT | The points S(i, j) with integer Cartesian coordinates 0 < i \leqn, |
| AUT2 | AUT | Let us denote by s(n) = |
| AUT3 | AUT | Let a, b, c be positive real numbers, 0 < a \leqb \leqc. Prove that |
| BUL4 | BUL | Let xn = 22n + 1 and let m be the least common multiple of |
| BUL6 | BUL | Let squares be constructed on the sides BC, CA, AB of a trian- |
| BUL7 | BUL | In a triangle ABC, let H be its orthocenter, O its circumcenter, |
| BUL9 | BUL | The base of an inclined prism is a triangle ABC. The per- |
| CUB10 | CUB | In how many different ways can three knights be placed on a |
| CUB11 | CUB | Prove that n! cannot be the square of any natural number. |
| CUB12 | CUB | A system of n numbers x1, x2, . . . , xn is given such that |
| CUB13 | CUB | One Martian, one Venusian, and one Human reside on Pluton. |
| GBR14 | GBR | Note that 83 −73 = 169 = 132 and 13 = 22 + 32. Prove that |
| GBR15 | GBR | Let ABCD be a convex quadrilateral whose diagonals intersect |
| GDR18 | GDR | Let a1, a2, . . . , an be positive numbers, mg = (a1a2 \cdot \cdot \cdot an)1/n |
| GDR19 | GDR | In a triangle P1P2P3 let PiQi be the altitude from Pi for |
| GDR20 | GDR | Let M be the circumcenter of a triangle ABC. The line through |
| HUN22 | HUN | We are given an n \times n board, where n is an odd number. In |
| HUN23 | HUN | Find all integer solutions of the equation |
| HUN24 | HUN | Let A, B, and C denote the angles of a triangle. If sin2 A + |
| HUN25 | HUN | Let ABC, AA1A2, BB1B2, CC1C2 be four equilateral triangles |
| HUN26 | HUN | An infinite set of rectangles in the Cartesian coordinate |
| NET29 | NET | A rhombus with its incircle is given. At each vertex of the |
| NET30 | NET | Prove that the system of equations |
| NET32 | NET | Two half-lines a and b, with the common endpoint O, make an |
| NET33 | NET | A square 2n \times 2n grid is given. Let us consider all possible |
| POL37 | POL | Let S be a circle, and \alpha = {A1, . . . , An} a family of open arcs |
| POL38 | POL | Let A, B, C be three points with integer coordinates in the |
| SWE40 | SWE | Prove that |
| SWE41 | SWE | Consider the set of grid points (m, n) in the plane, m, n inte- |
| SWE42 | SWE | Let Li, i = 1, 2, 3, be line segments on the sides of an equilateral |
| SWE43 | SWE | Show that for nonnegative real numbers a, b and integers n \geq2, |
| SWE45 | SWE | Let m and n denote integers greater than 1, and let \nu(n) be |
| USS48 | USS | A sequence of real numbers x1, x2, . . . , xn is given such that |
| USS49 | USS | Diagonals of a convex quadrilateral ABCD intersect at a |
| USS51 | USS | Suppose that the sides AB and DC of a convex quadrilateral |
| YUG53 | YUG | Denote by xn(p) the multiplicity of the prime p in the canonical |
| YUG54 | YUG | A set M is formed of |
| YUG55 | YUG | Prove that the polynomial x4 + \lambdax3 + µx2 + \nux + 1 has no |