IMO 1976 Longlist
IMO 1976 Longlist — 37 problems.
IMO 1976 Longlist
37 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| BUL2 | BUL | Let P be a set of n points and S a set of l segments. It is |
| BUL4 | BUL | Find all pairs of natural numbers (m, n) for which 2m \cdot 3n + 1 |
| BUL5 | BUL | Let ABCDS be a pyramid with four faces and with ABCD |
| CZS6 | CZS | For each point X of a given polytope, denote by f(X) the sum |
| CZS7 | CZS | Let P be a fixed point and T a given triangle that contains the |
| CZS9 | CZS | Find all (real) solutions of the system |
| FIN10 | FIN | Show that the reciprocal of any number of the form 2(m2 + |
| FIN12 | FIN | Five points lie on the surface of a ball of unit radius. Find the |
| GBR15 | GBR | Let ABC and A′B′C′ be any two coplanar triangles. Let L be |
| GBR16 | GBR | Prove that there is a positive integer n such that the decimal |
| GBR17 | GBR | Show that there exists a convex polyhedron with all its vertices |
| GDR18 | GDR | Prove that the number 191976 + 761976: |
| GDR19 | GDR | For a positive integer n, let 6(n) be the natural number whose |
| GDR20 | GDR | Let (an), n = 0, 1, . . ., be a sequence of real numbers such that |
| GDR21 | GDR | Find the largest positive real number p (if it exists) such that |
| GDR22 | GDR | A regular pentagon A1A2A3A4A5 with side length s is given. |
| NET23 | NET | Prove that in a Euclidean plane there are infinitely many |
| NET24 | NET | Let 0 \leqx1 \leqx2 \leq\cdot \cdot \cdot \leqxn \leq1. Prove that for all A… |
| NET27 | NET | In a plane three points P, Q, R, not on a line, are given. Let |
| POL30 | POL | Prove that if P(x) = (x−a)kQ(x), where k is a positive integer, |
| POL31 | POL | Into every lateral face of a quadrangular pyramid a circle is |
| POL32 | POL | We consider the infinite chessboard covering the whole plane. |
| SWE33 | SWE | A finite set of points P in the plane has the following prop- |
| SWE34 | SWE | Let {an}\infty |
| USA36 | USA | Three concentric circles with common center O are cut by a |
| USA37 | USA | From a square board 11 squares long and 11 squares wide, the |
| USA38 | USA | Let x = \sqrta + |
| USA39 | USA | In \triangleABC, the inscribed circle is tangent to side BC at X. |
| USA40 | USA | Let g(x) be a fixed polynomial and define f(x) by f(x) = |
| USS42 | USS | For a point O inside a triangle ABC, denote by A1, B1, C1 |
| USS43 | USS | Prove that if for a polynomial P(x, y) we have |
| USS44 | USS | A circle of radius 1 rolls around a circle of radius |
| USS45 | USS | We are given n (n \geq5) circles in a plane. Suppose that every |
| USS46 | USS | For a \geq0, b \geq0, c \geq0, d \geq0, prove the inequality |
| VIE49 | VIE | Determine whether there exist 1976 nonsimilar triangles with |
| VIE50 | VIE | Find a function f(x) defined for all real values of x such that |
| YUG51 | YUG | Four swallows are catching a fly. At first, the swallows are |
IMO 1976 LL NET24
Let 0 \leqx1 \leqx2 \leq\cdot \cdot \cdot \leqxn \leq1. Prove that for all A \geq1