IMO 1995 Shortlist
28 problems · Source: IMO Compendium
Algebra
| # |
Origin |
Problem |
| A1 |
RUS |
Let a, b, and c be positive real numbers such that abc = 1. |
| A2 |
SWE |
Let a and b be nonnegative integers such that ab \geqc2, |
| A3 |
UKR |
Let n be an integer, n \geq3. Let a1, a2, . . . , an be real numbers |
| A4 |
USA |
Let a, b, and c be given positive real numbers. Determine all |
| A5 |
UKR |
Let R be the set of real numbers. Does there exist a function |
| A6 |
JAP |
Let n be an integer, n \geq3. Let x1, x2, . . . , xn be real numbers |
Geometry
| # |
Origin |
Problem |
| G1 |
BUL |
Let A, B, C, and D be distinct points on a line, in that |
| G2 |
GER |
Let A, B, and C be noncollinear points. Prove that there is |
| G3 |
TUR |
The incircle of ABC touches BC, CA, and AB at D, E, and |
| G4 |
UKR |
An acute triangle ABC is given. Points A1 and A2 are taken |
| G5 |
NZL |
Let ABCDEF be a convex hexagon with AB = BC = |
| G6 |
USA |
Let A1A2A3A4 be a tetrahedron, G its centroid, and |
| G7 |
LAT |
O is a point inside a convex quadrilateral ABCD of area |
| G8 |
COL |
Let ABC be a triangle. A circle passing through B and C in- |
Number Theory
| # |
Origin |
Problem |
| N1 |
ROM |
Let k be a positive integer. Prove that there are infinitely |
| N2 |
RUS |
Let Z denote the set of all integers. Prove that for any integers |
| N3 |
CZE |
Determine all integers n > 3 such that there are n points |
| N4 |
BUL |
Find all positive integers x and y such that x+y2+z3 = xyz, |
| N5 |
IRE |
At a meeting of 12k people, each person exchanges greetings |
| N6 |
POL |
Let p be an odd prime. Find the number of p-element |
| N7 |
BLR |
Does there exist an integer n > 1 that satisfies the following |
| N8 |
GER |
Let p be an odd prime. Determine positive integers x and |
Problems
| # |
Origin |
Problem |
| 23 |
— |
S1 (UKR) Does there exist a sequence F(1), F(2), F(3), . . . of nonneg- |
| 24 |
— |
S2 (POL)IMO4 The positive real numbers x0, x1, . . . , x1995 satisfy x0 = |
| 25 |
— |
S3 (POL) For an integer x \geq1, let p(x) be the least prime that does not |
| 26 |
— |
S4 (NZL) Suppose that x1, x2, x3, . . . are positive real numbers for which |
| 27 |
— |
S5 (FIN) For positive integers n, the numbers f(n) are defined induc- |
| 28 |
— |
S6 (IND) Let N denote the set of all positive integers. Prove that there |