IMO 1988 Longlist
IMO 1988 Longlist — 63 problems.
IMO 1988 Longlist
63 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| BUL2 | BUL | Let an = |
| CUB5 | CUB | Let k be a positive integer and Mk the set of all the integers |
| FRA9 | FRA | If a0 is a positive real number, consider the sequence {an} |
| FRA12 | FRA | Show that there do not exist more than 27 half-lines (or rays) |
| FRA13 | FRA | Let T be a triangle with inscribed circle C. A square with sides |
| FRG15 | FRG | Let 1 \leqk < n. Consider all finite sequences of positive integers |
| FRG16 | FRG | Show that if n runs through all positive integers, f(n) = |
| FRG17 | FRG | Show that if n runs through all positive integers, f(n) = |
| GBR20 | GBR | It is proposed to partition the set of positive integers into two |
| GDR24 | GDR | Let Zm,n be the set of all ordered pairs (i, j) with i \in |
| GRE26 | GRE | Let AB and CD be two perpendicular chords of a circle with |
| GRE29 | GRE | Find positive integers x1, x2, . . . , x29, at least one of which is |
| HKG30 | HKG | Find the total number of different integers that the function |
| HKG31 | HKG | The circle x2 + y2 = r2 meets the coordinate axes at A = |
| HKG32 | HKG | Assuming that the roots of x3+px2+qx+r = 0 are all real and |
| HKG33 | HKG | Find a necessary and sufficient condition on the natural num- |
| HKG34 | HKG | Express the number 1988 as the sum of some positive integers |
| HKG35 | HKG | In the triangle ABC, let D, E, and F be the midpoints of the |
| HUN37 | HUN | Let n points be given on the surface of a sphere. Show that the |
| HUN38 | HUN | In a multiple choice test there were 4 questions and 3 possible |
| ICE40 | ICE | A sequence of numbers an, n = 1, 2, . . ., is defined as follows: |
| INA41 | INA | (a) Let ABC be a triangle with AB = 12 and AC = 16. Suppose M is the |
| INA42 | INA | (a) Four balls of radius 1 are mutually tangent, three resting an the floor |
| INA43 | INA | (a) The polynomial x2k + 1 + (x+ 1)2k is not divisible by x2 +x+ 1. Find |
| INA44 | INA | (a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the |
| INA45 | INA | (a) Consider a circle K with diameter AB, a circle L tangent to AB and |
| INA46 | INA | (a) Calculate x = (11+6 |
| IRE48 | IRE | Find all plane triangles whose sides have integer length and |
| IRE49 | IRE | Let −1 < x < 1. Show that |
| IRE50 | IRE | Let g(n) be defined as follows: |
| ISR51 | ISR | Let A1, A2, . . . , A29 be 29 different sequences of positive integers. |
| KOR53 | KOR | Let x = p, y = q, z = r, w = s be the unique solution of the |
| KOR55 | KOR | Find all positive integers x such that the product of all digits |
| KOR56 | KOR | The Fibonacci sequence is defined by |
| KOR57 | KOR | Let C be a cube with edges of length 2. Construct a solid with |
| KOR58 | KOR | For each pair of positive integers k and n, let Sk(n) be the |
| MEX61 | MEX | Prove that the numbers A, B, and C are equal, where we |
| MON62 | MON | The positive integer n has the property that in any set of n |
| MON63 | MON | Let ABCD be a quadrilateral. Let A′BCD′ be the reflection |
| MON64 | MON | Given n points A1, A2, . . . , An, no three collinear, show that |
| MON66 | MON | Suppose \alphai > 0, \betai > 0 for 1 \leqi \leqn (n > 1) and that |
| NET67 | NET | Given a set of 1988 points in the plane, no three points of the |
| NET68 | NET | Let S be the set of all sequences {ai | 1 \leqi \leq7, ai = 0 or 1}. |
| POL69 | POL | For a convex polygon P in the plane let P ′ denote the convex |
| POL70 | POL | In 3-dimensional space a point O is given and a finite set A |
| POL71 | POL | Given integers a1, . . . , a10, prove that there exists a nonzero |
| SIN73 | SIN | In a group of n people each one knows exactly three others. They |
| SPA75 | SPA | Let ABC be a triangle with inradius r and circumradius R. |
| SPA76 | SPA | The quadrilateral A1A2A3A4 is cyclic and its sides are a1 = |
| SPA77 | SPA | Consider h + 1 chessboards. Number the squares of each board |
| SWE78 | SWE | A two-person game is played with nine boxes arranged in a |
| SWE80 | SWE | Let S be an infinite set of integers containing zero and such |
| USA81 | USA | There are n \geq3 job openings at a factory, ranked 1 to n in |
| USA82 | USA | The triangle ABC has a right angle at C. The point P is |
| USS86 | USS | Let a, b, c be integers different from zero. It is known that the |
| USS87 | USS | All the irreducible positive rational numbers such that the prod- |
| USS88 | USS | There are six circles inside a fixed circle, each tangent to |
| VIE89 | VIE | We match sets M of points in the coordinate plane to sets M∗ |
| VIE90 | VIE | Does there exist a number \alpha (0 < \alpha < 1) such that there is an |
| VIE91 | VIE | A regular 14-gon with side length a is inscribed in a circle of |
| VIE92 | VIE | Let p \geq2 be a natural number. Prove that there exists an |
| VIE93 | VIE | Given a natural number n, find all polynomials P(x) of degree |
| VIE94 | VIE | Let n + 1 (n \geq1) positive integers be given such that for each |