IMO 1978 Longlist
IMO 1978 Longlist — 37 problems.
IMO 1978 Longlist
37 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| BUL2 | BUL | If |
| BUL3 | BUL | Find all numbers \alpha for which the equation |
| CUB5 | CUB | Prove that for any triangle ABC there exists a point P in the |
| CUB6 | CUB | Prove that for all X > 1 there exists a triangle whose sides |
| CZS8 | CZS | For two given triangles A1A2A3 and B1B2B3 with areas ∆A |
| CZS10 | CZS | Show that for any natural number n there exist two prime |
| CZS11 | CZS | Find all natural numbers n < 1978 with the following property: |
| FIN12 | FIN | The equation x3 + ax2 + bx + c = 0 has three (not necessarily |
| FIN13 | FIN | The satellites A and B circle the Earth in the equatorial plane |
| FIN14 | FIN | Let p(x, y) and q(x, y) be polynomials in two variables such |
| FRA15 | FRA | Prove that for every positive integer n coprime to 10 there |
| FRA18 | FRA | Given a natural number n, prove that the number M(n) of |
| GBR20 | GBR | Let O be the center of a circle. Let OU, OV be perpendicular |
| GBR21 | GBR | A circle touches the sides AB, BC, CD, DA of a square at |
| GBR22 | GBR | Two nonzero integers x, y (not necessarily positive) are such |
| GDR25 | GDR | Consider a polynomial P(x) = ax2 + bx + c with a > 0 that |
| GDR27 | GDR | Determine the sixth number after the decimal point in the |
| GDR28 | GDR | Let c, s be real functions defined on R{0} that are nonconstant |
| GDR29 | GDR | (Variant of GDR 4) Given a nonconstant function f : R+ \toR |
| NET31 | NET | Let the polynomials |
| NET32 | NET | Let C be the circumcircle of the square with vertices (0, 0), |
| SWE33 | SWE | A sequence (an)\infty |
| SWE35 | SWE | A sequence (an)N |
| TUR36 | TUR | The integers 1 through 1000 are located on the circumference |
| TUR37 | TUR | Simplify |
| TUR38 | TUR | Given a circle, construct a chord that is trisected by two given |
| TUR39 | TUR | A is a 2m-digit positive integer each of whose digits is 1. B is |
| TUR40 | TUR | If Cp |
| USA42 | USA | A, B, C, D, E are points on a circle O with radius equal to r. |
| USA43 | USA | If p is a prime greater than 3, show that at least one of the |
| USA44 | USA | In \triangleABC with \angleC = 60o, prove that c |
| USA45 | USA | If r > s > 0 and a > b > c, prove that |
| VIE47 | VIE | Given the expression |
| VIE49 | VIE | Let A, B, C, D be four arbitrary distinct points in space. |
| VIE50 | VIE | A variable tetrahedron ABCD has the following properties: |
| VIE51 | VIE | Find the relations among the angles of the triangle ABC whose |
| YUG54 | YUG | Let p, q and r be three lines in space such that there is no plane |