IMO 1970 Longlist
IMO 1970 Longlist — 47 problems.
IMO 1970 Longlist
47 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUT1 | AUT | Prove that |
| AUT2 | AUT | Prove that the two last digits of 999 and 9999 |
| AUT3 | AUT | Prove that for a, b \inN, a!b! divides (a + b)!. |
| AUT4 | AUT | Solve the system of equations |
| AUT5 | AUT | Prove that |
| BEL6 | BEL | Prove that the equation in x |
| BEL7 | BEL | Let ABCD be any quadrilateral. A square is constructed on |
| BEL9 | BEL | If n is even, prove that |
| BEL10 | BEL | Let A, B, C be angles of a triangle. Prove that |
| BEL11 | BEL | Let ABCD and A′B′C′D′ be two squares in the same plane and |
| BUL12 | BUL | Let x1, x2, x3, x4, x5, x6 be given integers, not divisible by 7. |
| BUL13 | BUL | A triangle ABC is given. Each side of ABC is divided into equal |
| BUL14 | BUL | Let \alpha + \beta + \gamma = \pi. Prove that |
| BUL15 | BUL | Given a triangle ABC, let R be the radius of its circumcir- |
| BUL16 | BUL | Show that the equation |
| CZS19 | CZS | Let n > 1 be a natural number, a \geq1 a real number, and |
| CZS21 | CZS | Find necessary and sufficient conditions on given positive num- |
| FRA23 | FRA | Let E be a finite set, PE the family of its subsets, and f a |
| FRA24 | FRA | Let n and p be two integers such that 2p \leqn. Prove the |
| FRA25 | FRA | Suppose that f is a real function defined for 0 \leqx \leq1 having |
| FRA26 | FRA | Consider a finite set of vectors in space {a1, a2, . . . , an} and |
| FRA27 | FRA | Find a natural number n such that for all prime numbers p, n |
| GDR28 | GDR | A set G with elements u, v, w, . . . is a group if the following |
| GDR29 | GDR | Prove that the equation 4x+6x = 9x has no rational solutions. |
| GDR31 | GDR | Prove that for any triangle with sides a, b, c and area P the |
| NET32 | NET | Let there be given an acute angle \angleAOB = 3\alpha, where OA = |
| NET33 | NET | The vertices of a given square are clockwise lettered A, B, C, D. |
| NET34 | NET | In connection with a convex pentagon ABCDE we consider |
| NET35 | NET | Find for every value of n a set of numbers p for which the fol- |
| NET36 | NET | Let x, y, z be nonnegative real numbers satisfying |
| NET37 | NET | Solve the set of simultaneous equations |
| POL38 | POL | Find the greatest integer A for which in any permutation of |
| POL40 | POL | Let ABC be a triangle with angles \alpha, \beta, \gamma commensurable with |
| POL41 | POL | Let a cube of side 1 be given. Prove that there exists a point |
| ROM43 | ROM | Prove that the equation |
| ROM44 | ROM | If a, b, c are side lengths of a triangle, prove that |
| ROM45 | ROM | Let M be an interior point of tetrahedron V ABC. Denote |
| ROM46 | ROM | Given a triangle ABC and a plane \pi having no common points |
| ROM47 | ROM | Given a polynomial |
| ROM48 | ROM | Let a polynomial p(x) with integer coefficients take the value |
| SWE49 | SWE | For n \inN, let f(n) be the number of positive integers k \leqn |
| SWE50 | SWE | The area of a triangle is S and the sum of the lengths of its |
| SWE51 | SWE | Let p be a prime number. A rational number x, with 0 < x < 1, |
| SWE53 | SWE | A square ABCD is divided into (n −1)2 congruent squares, |
| USS55 | USS | A turtle runs away from an UFO with a speed of 0.2 m/s. The |
| USS56 | USS | A square hole of depth h whose base is of length a is given. |
| USS57 | USS | Let the numbers 1, 2, . . . , n2 be written in the cells of an n \times n |