IMO 1999 Shortlist
27 problems · Source: IMO Compendium
Algebra
| # |
Origin |
Problem |
| A1 |
POL |
Let n \geq2 be a fixed integer. Find the least constant C |
| A2 |
RUS |
The numbers from 1 to n2 are randomly arranged in the cells |
| A3 |
FIN |
A game is played by n girls (n \geq2), everybody having a ball. |
| A4 |
BLR |
Prove that the set of positive integers cannot be partitioned |
| A5 |
JAP |
Find all the functions f : R \toR that satisfy |
| A6 |
SWE |
For n \geq3 and a1 \leqa2 \leq\cdot \cdot \cdot \leqan given real numbers we |
Combinatorics
| # |
Origin |
Problem |
| C1 |
IND |
Let n \geq1 be an integer. A path from (0, 0) to (n, n) in the |
| C2 |
CAN |
(a) If a 5 \times n rectangle can be tiled using n pieces like those |
| C3 |
GBR |
A biologist watches a chameleon. The chameleon catches |
| C4 |
GBR |
Let A be a set of N residues (mod N 2). Prove that there |
| C5 |
BLR |
Let n be an even positive integer. We say that two dif- |
| C6 |
GBR |
Suppose that every integer has been given one of the colors |
| C7 |
IRE |
Let p > 3 be a prime number. For each nonempty subset T of |
Geometry
| # |
Origin |
Problem |
| G1 |
ARM |
Let ABC be a triangle and M an interior point. Prove that |
| G2 |
JAP |
A circle is called a separator for a set of five points in a plane |
| G3 |
EST |
A set S of points in space will be called completely sym- |
| G4 |
GBR |
For a triangle T = ABC we take the point X on the side |
| G5 |
FRA |
Let ABC be a triangle, Ωits incircle and Ωa, Ωb, Ωc three |
| G6 |
RUS |
Two circles Ω1 and Ω2 touch internally the circle Ωin |
| G7 |
ARM |
The point M inside the convex quadrilateral ABCD is such |
| G8 |
RUS |
Points A, B, C divide the circumcircle Ωof the triangle ABC |
Number Theory
| # |
Origin |
Problem |
| N1 |
TWN |
Find all pairs of positive integers (x, p) such that p is |
| N2 |
ARM |
Prove that every positive rational number can be repre- |
| N3 |
RUS |
Prove that there exist two strictly increasing sequences (an) |
| N4 |
FRA |
Denote by S the set of all primes p such that the decimal |
| N5 |
ARM |
Let n, k be positive integers such that n is not divisible by |
| N6 |
BLR |
Prove that for every real number M there exists an infinite |