IMO 1969 Longlist
IMO 1969 Longlist — 71 problems.
IMO 1969 Longlist
71 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| BEL1 | BEL | A parabola P1 with equation x2 −2py = 0 and parabola P2 |
| BEL2 | BEL | (a) Find the equations of regular hyperbolas passing through |
| BEL3 | BEL | Construct the circle that is tangent to three given circles. |
| BEL4 | BEL | Let O be a point on a nondegenerate conic. A right angle with |
| BEL5 | BEL | Let G be the centroid of the triangle OAB. |
| BEL6 | BEL | Evaluate (cos(\pi/4) + i sin(\pi/4))10 in two different ways and |
| BUL7 | BUL | Prove that the equation |
| BUL8 | BUL | Find all functions f defined for all x that satisfy the condition |
| BUL9 | BUL | One hundred convex polygons are placed on a square with edge |
| BUL10 | BUL | Let M be the point inside the right-angled triangle ABC |
| BUL11 | BUL | Let Z be a set of points in the plane. Suppose that there exists |
| CZS12 | CZS | Given a unit cube, find the locus of the centroids of all tetra- |
| CZS13 | CZS | Let p be a prime odd number. Is it possible to find p−1 natural |
| CZS14 | CZS | Let a and b be two positive real numbers. If x is a real solution |
| CZS15 | CZS | Let K1, . . . , Kn be nonnegative integers. Prove that |
| CZS16 | CZS | A convex quadrilateral ABCD with sides AB = a, BC = b, |
| CZS17 | CZS | Let d and p be two real numbers. Find the first term of an arith- |
| FRA18 | FRA | Let a and b be two nonnegative integers. Denote by H(a, b) |
| FRA19 | FRA | Let n be an integer that is not divisible by any square greater |
| FRA20 | FRA | A polygon (not necessarily convex) with vertices in the lattice |
| FRA21 | FRA | A right-angled triangle OAB has its right angle at the point B. |
| FRA22 | FRA | Let \alpha(n) be the number of pairs (x, y) of integers such that |
| FRA23 | FRA | Consider the integer d = ab−1 |
| GBR24 | GBR | The polynomial P(x) = a0xk + a1xk−1 + \cdot \cdot \cdot + ak, where |
| GBR25 | GBR | Let a, b, x, y be positive integers such that a and b have no |
| GBR26 | GBR | A smooth solid consists of a right circular cylinder of height |
| GBR27 | GBR | The segment AB perpendicularly bisects CD at X. Show that, |
| GBR28 | GBR | Let us define u0 = 0, u1 = 1 and for n \geq0, un+2 = aun+1+bun, |
| GDR29 | GDR | Find all real numbers \lambda such that the equation |
| GDR30 | GDR | Prove that there exist infinitely many natural numbers a |
| GDR31 | GDR | Find the number of permutations a1, . . . , an of the set |
| GDR32 | GDR | Find the maximal number of regions into which a sphere can |
| GDR33 | GDR | Given a ring G in the plane bounded by two concentric circles |
| HUN34 | HUN | Let a and b be arbitrary integers. Prove that if k is an integer |
| HUN35 | HUN | Prove that |
| HUN36 | HUN | In the plane 4000 points are given such that each line passes |
| HUN37 | HUN | If a1, a2, . . . , an are real constants, and if |
| HUN38 | HUN | Let r and m (r \leqm) be natural numbers and Ak = 2k−1 |
| HUN39 | HUN | Find the positions of three points A, B, C on the boundary of |
| MON40 | MON | Find the number of five-digit numbers with the following |
| MON41 | MON | Given two numbers x0 and x1, let \alpha and \beta be coefficients |
| MON42 | MON | Let Ak (1 \leqk \leqh) be n-element sets such that each two |
| MON43 | MON | Let p and q be two prime numbers greater than 3. Prove that |
| MON44 | MON | Find the radius of the circle circumscribed about the isosceles |
| MON45 | MON | Given n points in the plane such that no three of them |
| NET46 | NET | The vertices of an (n + 1)-gon are placed on the edges of a |
| NET47 | NET | Let A and B be points on the circle \gamma. A point C, different |
| NET48 | NET | Let x1, x2, x3, x4, and x5 be positive integers satisfying |
| NET49 | NET | A boy has a set of trains and pieces of railroad track. Each |
| NET50 | NET | The bisectors of the exterior angles of a pentagon B1B2B3B4B5 |
| NET51 | NET | A curve determined by |
| POL52 | POL | Prove that a regular polygon with an odd number of edges |
| POL53 | POL | Given two segments AB and CD not in the same plane, find |
| POL54 | POL | Given a polynomial f(x) with integer coefficients whose value |
| POL55 | POL | Find the conditions on the positive real number a such that |
| POL56 | POL | Let a and b be two natural numbers that have an equal number |
| POL57 | POL | On the sides AB and AC of triangle ABC two points K and |
| SWE58 | SWE | Six points P1, . . . , P6 are given in 3-dimensional space such that |
| SWE59 | SWE | For each \lambda (0 < \lambda < 1 and \lambda ̸= 1/n for all n = 1, 2, 3, . . .) |
| SWE60 | SWE | Find the natural number n with the following properties: |
| SWE61 | SWE | Let a0, a1, a2 be determined with a0 = 0, an+1 = 2an + 2n. |
| SWE62 | SWE | Which natural numbers can be expressed as the difference of |
| SWE63 | SWE | Prove that there are infinitely many positive integers that |
| USS64 | USS | Prove that for a natural number n > 2, |
| USS65 | USS | Prove that for a > b2, |
| USS66 | USS | (a) Prove that if 0 \leqa0 \leqa1 \leqa2, then |
| USS67 | USS | Under the conditions x1, x2 > 0, x1y1 > z2 |
| USS68 | USS | Given 5 points in the plane, no three of which are collinear, prove |
| YUG69 | YUG | Suppose that positive real numbers x1, x2, x3 satisfy |
| YUG70 | YUG | A park has the shape of a convex pentagon of area 5 |
| YUG71 | YUG | Let four points Ai (i = 1, 2, 3, 4) in the plane determine four |