IMO 1977 Longlist
IMO 1977 Longlist — 44 problems.
IMO 1977 Longlist
44 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| BUL1 | BUL | A pentagon ABCDE inscribed in a circle for which BC < CD |
| BUL3 | BUL | In a company of n persons, each person has no more than d |
| BUL4 | BUL | We are given n points in space. Some pairs of these points |
| CZS6 | CZS | Let x1, x2, . . . , xn (n \geq1) be real numbers such that 0 \leqxj \leq\pi, |
| CZS7 | CZS | Prove the following assertion: If c1, c2, . . . , cn (n \geq2) are real |
| CZS8 | CZS | A hexahedron ABCDE is made of two regular congruent tetra- |
| CZS9 | CZS | Let ABCD be a regular tetrahedron and Z an isometry map- |
| FIN43 | FIN | Evaluate |
| FIN44 | FIN | Let E be a finite set of points in space such that E is not |
| FIN46 | FIN | Let f be a strictly increasing function defined on the set of real |
| FRG11 | FRG | Let n and z be integers greater than 1 and (n, z) = 1. Prove: |
| FRG12 | FRG | Let z be an integer > 1 and let M be the set of all numbers |
| GBR19 | GBR | Given any integer m > 1 prove that there exist infinitely |
| GBR21 | GBR | Given that x1+x2+x3 = y1+y2+y3 = x1y1+x2y2+x3y3 = 0, |
| GDR15 | GDR | Let n be an integer greater than 1. In the Cartesian coordinate |
| GDR17 | GDR | A ball K of radius r is touched from the outside by mutually |
| GDR18 | GDR | Given an isosceles triangle ABC with a right angle at C, |
| HUN24 | HUN | Determine all real functions f(x) that are defined and contin- |
| HUN25 | HUN | Prove the identity |
| NET26 | NET | Let p be a prime number greater than 5. Let V be the collection |
| NET30 | NET | A triangle ABC with \angleA = 30◦and \angleC = 54◦is given. On |
| POL31 | POL | Let f be a function defined on the set of pairs of nonzero |
| POL32 | POL | In a room there are nine men. Among every three of them there |
| POL33 | POL | A circle K centered at (0, 0) is given. Prove that for every vector |
| ROM35 | ROM | Find all numbers N = a1a2 . . . an for which 9 \times a1a2 . . . an = |
| ROM36 | ROM | Consider a sequence of numbers (a1, a2, . . . , a2n). Define the |
| ROM37 | ROM | Let A1, A2, . . . , An+1 be positive integers such that (Ai, An+1) |
| ROM38 | ROM | Let mj > 0 for j = 1, 2, . . ., n and a1 \leq\cdot \cdot \cdot \leqan < b1… |
| ROM39 | ROM | Consider 37 distinct points in space, all with integer coordi- |
| SWE40 | SWE | The numbers 1, 2, 3, . . ., 64 are placed on a chessboard, one |
| SWE41 | SWE | A wheel consists of a fixed circular disk and a mobile circular |
| SWE42 | SWE | The sequence an,k, k = 1, 2, 3, . . ., 2n, n = 0, 1, 2, . . ., is defined |
| USA52 | USA | Two perpendicular chords are drawn through a given interior |
| USA53 | USA | Find all pairs of integers a and b for which |
| USA54 | USA | If 0 \leqa \leqb \leqc \leqd, prove that |
| USA55 | USA | Through a point O on the diagonal BD of a parallelogram |
| USA56 | USA | The four circumcircles of the four faces of a tetrahedron have |
| USS47 | USS | A square ABCD is given. A line passing through A intersects |
| USS48 | USS | The intersection of a plane with a regular tetrahedron with |
| USS49 | USS | Find all pairs of integers (p, q) for which all roots of the trino- |
| USS50 | USS | Determine all positive integers n for which there exists a poly- |
| USS51 | USS | Several segments, which we shall call white, are given, and |
| VIE58 | VIE | Prove that for every triangle the following inequality holds: |
| VIE60 | VIE | Suppose x0, x1, . . . , xn are integers and x0 > x1 > \cdot \cdot \cdot > xn. |