IMO 1992 Longlist
IMO 1992 Longlist — 61 problems.
IMO 1992 Longlist
61 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUS1 | AUS | Points D and E are chosen on the sides AB and AC of the |
| AUS3 | AUS | Let ABC be a triangle, O its circumcenter, S its centroid, and |
| CAN4 | CAN | Let p, q, and r be the angles of a triangle, and let a = sin 2p, |
| CAN5 | CAN | Let I, H, O be the incenter, centroid, and circumcenter of the |
| CAN6 | CAN | Suppose that n numbers x1, x2, . . . , xn are chosen randomly |
| CAN7 | CAN | Let X be a bounded, nonempty set of points in the Cartesian |
| COL11 | COL | Let \varphi(n, m), m ̸= 1, be the number of positive integers less |
| COL12 | COL | Given a triangle ABC such that the circumcenter is in the |
| FIN14 | FIN | Integers a1, a2, . . . , an satisfy |ak| = 1 and |
| FIN15 | FIN | Prove that there exist 78 lines in the plane such that they have |
| FIN16 | FIN | Find all triples (x, y, z) of integers such that |
| FRG18 | FRG | Fibonacci numbers are defined as follows: F1 = F2 = 1, Fn+2 = |
| FRG19 | FRG | Denote by an the greatest number that is not divisible by 3 |
| FRG20 | FRG | Let X and Y be two sets of points in the plane and M be a set |
| GBR21 | GBR | Prove that if x, y, z > 1 and 1 |
| HKG23 | HKG | An Egyptian number is a positive integer that can be expressed |
| ICE24 | ICE | Let Q+ denote the set of nonnegative rational numbers. Show |
| IND25 | IND | (a) Show that the set N of all natural numbers can be parti- |
| IND27 | IND | Let ABC be an arbitrary scalene triangle. Define \Sigma to be the |
| IND30 | IND | Let Pn = (19 + 92)(192 + 922) \cdot \cdot \cdot (19n + 92n) for each positive |
| IRE32 | IRE | Let Sn = {1, 2, . . ., n} and fn : Sn \toSn be defined inductively |
| IRE33 | IRE | Let a, b, c be positive real numbers and p, q, r complex numbers. |
| IRE34 | IRE | Let a, b, c be integers. Prove that there are integers p1, q1, r1, |
| IRN36 | IRN | Find all rational solutions of |
| IRN37 | IRN | Let the circles C1, C2, and C3 be orthogonal to the circle C |
| ITA39 | ITA | Let n \geq2 be an integer. Find the minimum k for which there |
| ITA40 | ITA | The colonizers of a spherical planet have decided to build N |
| JAP41 | JAP | Let S be a set of positive integers n1, n2, . . . , n6 and let n(f) |
| KOR43 | KOR | Find the number of positive integers n satisfying \varphi(n) | n such |
| KOR45 | KOR | Let n be a positive integer. Prove that the number of ways |
| KOR46 | KOR | Prove that the sequence 5, 12, 19, 26, 33, . . . contains no term |
| KOR47 | KOR | Find the largest integer not exceeding $1992 |
| MON48 | MON | Find all the functions f : R+ \toR satisfying the identity |
| MON49 | MON | Given real numbers xi (i = 1, 2, . . . , 4x + 2) such that |
| MON50 | MON | Let N be a point inside the triangle ABC. Through the mid- |
| NET52 | NET | Let n be an integer > 1. In a circular arrangement of n lamps |
| POL54 | POL | Suppose that n > m \geq1 are integers such that the string of |
| POL56 | POL | A directed graph (any two distinct vertices joined by at most |
| POL57 | POL | For positive numbers a, b, c define A = (a + b + c)/3, G = |
| POR58 | POR | Let ABC be a triangle. Denote by a, b, and c the lengths of |
| PRK59 | PRK | Let a regular 7-gon A0A1A2A3A4A5A6 be inscribed in a circle. |
| PRK61 | PRK | There are a board with 2n\cdot2n (= 4n2) squares and 4n2−1 cards |
| ROM62 | ROM | Let c1, . . . , cn (n \geq2) be real numbers such that 0 \leq ci \leqn. |
| ROM63 | ROM | Let a and b be integers. Prove that 2a2−1 |
| ROM64 | ROM | For any positive integer n consider all representations n = |
| SAF65 | SAF | If A, B, C, and D are four distinct points in space, prove that |
| SPA66 | SPA | A circle of radius \rho is tangent to the sides AB and AC of the |
| SPA67 | SPA | In a triangle, a symmedian is a line through a vertex that is |
| SPA68 | SPA | Show that the numbers tan(r\pi/15), where r is a positive integer |
| THA70 | THA | Let two circles A and B with unequal radii r and R, respec- |
| THA71 | THA | Let P1(x, y) and P2(x, y) be two relatively prime polynomials |
| TUR72 | TUR | In a school six different courses are taught: mathematics, |
| TUR73 | TUR | Let {An | n = 1, 2, . . .} be a set of points in the plane such |
| TUR74 | TUR | Let S = |
| TWN75 | TWN | A sequence {an} of positive integers is defined by |
| TWN76 | TWN | Given any triangle ABC and any positive integer n, we say |
| TWN77 | TWN | Show that if 994 integers are chosen from 1, 2, . . . , 1992 and |
| USA78 | USA | Let Fn be the nth Fibonacci number, defined by F1 = F2 = 1 |
| USA80 | USA | Given a graph with n vertices and a positive integer m that is |
| USA81 | USA | Suppose that points X, Y, Z are located on sides BC, CA, |
| VIE82 | VIE | Let f(x) = xm + a1xm−1 + \cdot \cdot \cdot + am−1x + am and g(x) = |