IMO 1972 Longlist
IMO 1972 Longlist — 34 problems.
IMO 1972 Longlist
34 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| BUL1 | BUL | Find all integer solutions of the equation |
| BUL2 | BUL | Find all real values of the parameter a for which the system of |
| BUL3 | BUL | On a line a set of segments is given of total length less than |
| BUL4 | BUL | Given a triangle, prove that the points of intersection of three |
| BUL5 | BUL | Given a pyramid whose base is an n-gon inscribable in a circle, |
| BUL6 | BUL | Prove the inequality |
| CZS9 | CZS | Given natural numbers k and n, k \leqn, n \geq3, find the set |
| CZS10 | CZS | Given five points in the plane, no three of which are collinear, |
| CZS12 | CZS | A circle k = (S, r) is given and a hexagon AA′BB′CC′ inscribed |
| CZS13 | CZS | Given a sphere K, determine the set of all points A that are |
| GBR16 | GBR | Consider the set S of all the different odd positive integers |
| GBR17 | GBR | A solid right circular cylinder with height h and base-radius |
| GBR18 | GBR | We have p players participating in a tournament, each player |
| GBR19 | GBR | Let S be a subset of the real numbers with the following |
| MON23 | MON | Does there exist a 2n-digit number a2na2n−1 . . . a1 (for an |
| MON24 | MON | The diagonals of a convex 18-gon are colored in 5 different |
| NET25 | NET | We consider n real variables xi (1 \leqi \leqn), where n is an |
| NET28 | NET | The lengths of the sides of a rectangle are given to be odd |
| NET29 | NET | Let A, B, C be points on the sides B1C1, C1A1, A1B1 of a |
| ROM31 | ROM | Find values of n \inN for which the fraction 3n−2 |
| ROM32 | ROM | If n1, n2, . . . , nk are natural numbers and n1+n2+\cdot \cdot \cdot+nk = n, |
| ROM33 | ROM | A rectangle ABCD is given whose sides have lengths 3 and |
| ROM34 | ROM | If p is a prime number greater than 2 and a, b, c integers not |
| ROM35 | ROM | (a) Prove that for a, b, c, d \inR, m \in[1, +\infty) with am + b = |
| ROM36 | ROM | A finite number of parallel segments in the plane are given with |
| SWE37 | SWE | On a chessboard (8 \times 8 squares with sides of length 1) two |
| SWE38 | SWE | Congruent rectangles with sides m (cm) and n (cm) are |
| SWE39 | SWE | How many tangents to the curve y = x3 −3x (y = x3 + px) |
| SWE40 | SWE | Prove the inequalities |
| SWE41 | SWE | The ternary expansion x = 0.10101010 . . . is given. Give the |
| SWE42 | SWE | The decimal number 13101 is given. It is instead written as a |
| USS43 | USS | A fixed point A inside a circle is given. Consider all chords |
| USS45 | USS | Let ABCD be a convex quadrilateral whose diagonals AC and |
| USS46 | USS | Numbers 1, 2, . . . , 16 are written in a 4\times4 square matrix so that |