IMO 1979 Longlist
IMO 1979 Longlist — 55 problems.
IMO 1979 Longlist
55 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| BEL2 | BEL | For a finite set E of cardinality n \geq3, let f(n) denote the |
| BEL3 | BEL | Is it possible to partition 3-dimensional Euclidean space into |
| BEL5 | BEL | Describe which natural numbers do not belong to the set |
| BEL6 | BEL | Prove that 1 |
| BRA7 | BRA | M = (ai,j), i, j = 1, 2, 3, 4, is a square matrix of order four. |
| BRA8 | BRA | The sequence (an) of real numbers is defined as follows: |
| BRA9 | BRA | The real numbers \alpha1, \alpha2, \alpha3, . . . , \alphan are positive. Let… |
| BUL11 | BUL | Prove that a pyramid A1A2 . . . A2k+1S with equal lateral edges |
| BUL13 | BUL | The plane is divided into equal squares by parallel lines; i.e., |
| CZS14 | CZS | Let S be a set of n2 + 1 closed intervals (n a positive integer). |
| CZS16 | CZS | Let Q be a square with side length 6. Find the smallest integer |
| FIN18 | FIN | Show that for no integers a \geq1, n \geq1 is the sum |
| FIN19 | FIN | For k = 1, 2, . . . consider the k-tuples (a1, a2, . . . , ak) of positive |
| FRA21 | FRA | Let E be the set of all bijective mappings from R to R satisfying |
| FRA22 | FRA | Consider two quadrilaterals ABCD and A′B′C′D′ in an affine |
| FRA23 | FRA | Consider the set E consisting of pairs of integers (a, b), with a \geq |
| FRA24 | FRA | Let a and b be coprime integers, greater than or equal to 1. |
| FRG26 | FRG | Let n be a natural number. If 4n + 2n + 1 is a prime, prove |
| GDR30 | GDR | Let M be a set of points in a plane with at least two elements. |
| GDR32 | GDR | Let n, k \geq1 be natural numbers. Find the number A(n, k) of |
| GRE34 | GRE | Notice that in the fraction 16 |
| GRE35 | GRE | Given a sequence (an), with a1 = 4 and an+1 = a2 |
| GRE36 | GRE | A regular tetrahedron A1B1C1D1 is inscribed in a regular |
| HUN38 | HUN | Prove the following statement: If a polynomial f(x) with |
| HUN39 | HUN | A desert expedition camps at the border of the desert, and |
| HUN40 | HUN | A polynomial P(x) has degree at most 2k, where k = 0, 1, |
| HUN41 | HUN | Prove the following statement: There does not exist a pyramid |
| HUN42 | HUN | Let a quadratic polynomial g(x) = ax2 + bx + c be given and |
| ISR43 | ISR | Let a, b, c denote the lengths of the sides BC, CA, AB, respec- |
| ISR45 | ISR | For any positive integer n we denote by F(n) the number of |
| NET48 | NET | In the plane a circle C of unit radius is given. For any line l |
| NET49 | NET | Let there be given two sequences of integers fi(1), fi(2), . . . |
| POL51 | POL | Let ABC be an arbitrary triangle and let S1, S2, . . . , S7 be |
| POL52 | POL | Let a real number \lambda > 1 be given and a sequence (nk) of positive |
| POL53 | POL | An infinite increasing sequence of positive integers nj (j = |
| ROM55 | ROM | Let a, b be coprime integers. Show that the equation ax2 + |
| ROM56 | ROM | Show that for every natural number n, n |
| ROM57 | ROM | Let M be a set, and A, B, C given subsets of M. Find a |
| ROM58 | ROM | Prove that there exists a natural number k0 such that for |
| SWE59 | SWE | Determine the maximum value of x2y2z2w when x, y, z, w \geq0 |
| SWE61 | SWE | Let a1 \leqa2 \leq\cdot \cdot \cdot \leqan and b1 \leqb2 \leq\cdot \cdot \cdot… |
| SWE62 | SWE | T is a given triangle with vertices P1, P2, P3. Consider an arbi- |
| USA63 | USA | If a1, a2, . . . , an denote the lengths of the sides of an arbitrary |
| USA64 | USA | From point P on arc BC of the circumcircle about triangle |
| USA65 | USA | Given f(x) \leqx for all real x and |
| USS70 | USS | There are 1979 equilateral triangles: T1, T2, . . . , T1979. A side of |
| VIE72 | VIE | Let f(x) be a polynomial with integer coefficients. Prove that |
| VIE73 | VIE | In a plane a finite number of equal circles are given. These circles |
| VIE74 | VIE | Given an equilateral triangle ABC of side a in a plane, let |
| VIE75 | VIE | Given an equilateral triangle ABC, let M be an arbitrary point |
| VIE76 | VIE | Suppose that a triangle whose sides are of integer lengths is |
| YUG77 | YUG | By h(n), where n is an integer greater than 1, let us denote the |
| YUG78 | YUG | By \omega(n), where n is an integer greater than 1, let us denote |
| YUG79 | YUG | Let S be a unit circle and K a subset of S consisting of several |
| YUG81 | YUG | Let P be the set of rectangular parallelepipeds that have at |