IMO 2000 Shortlist
27 problems · Source: IMO Compendium
Algebra
| # |
Origin |
Problem |
| A1 |
USA |
Let a, b, c be positive real numbers with product 1. Prove |
| A2 |
GBR |
Let a, b, c be positive integers satisfying the conditions b > 2a |
| A3 |
BLR |
Find all pairs of functions f : R \toR, g : R \toR such that |
| A4 |
GBR |
The function F is defined on the set of nonnegative integers |
| A5 |
BLR |
Let n \geq2 be a positive integer and \lambda a positive real |
| A6 |
IRE |
A nonempty set A of real numbers is called a B3-set if the |
| A7 |
RUS |
For a polynomial P of degree 2000 with distinct real co- |
Combinatorics
| # |
Origin |
Problem |
| C1 |
HUN |
A magician has one hundred cards numbered 1 to 100. |
| C2 |
ITA |
A brick staircase with three steps of width 2 is made of twelve |
| C3 |
COL |
Let n \geq4 be a fixed positive integer. Given a set S = |
| C4 |
CZE |
Let n and k be positive integers such that n/2 < k \leq2n/3. |
| C5 |
RUS |
In the plane we have n rectangles with parallel sides. The |
| C6 |
FRA |
Let p and q be relatively prime positive integers. A subset |
Geometry
| # |
Origin |
Problem |
| G1 |
NET |
In the plane we are given two circles intersecting at X and Y . |
| G2 |
RUS |
Two circles G1 and G2 intersect at M and N. Let AB |
| G3 |
IND |
Let O be the circumcenter and H the orthocenter of an acute |
| G4 |
RUS |
Let A1A2 . . . An be a convex polygon, n \geq4. Prove that |
| G5 |
GBR |
The tangents at B and A to the circumcircle of an acute- |
| G6 |
ARG |
Let ABCD be a convex quadrilateral with AB not parallel |
| G7 |
IRN |
Ten gangsters are standing on a flat surface, and the distances |
| G8 |
RUS |
A1A2A3 is an acute-angled triangle. The foot of the |
Number Theory
| # |
Origin |
Problem |
| N1 |
JAP |
Determine all positive integers n \geq2 that satisfy the following |
| N2 |
FRA |
For a positive integer n, let d(n) be the number of all positive |
| N3 |
RUS |
Does there exist a positive integer n such that n has |
| N4 |
BRA |
Determine all triples of positive integers (a, m, n) such that |
| N5 |
BUL |
Prove that there exist infinitely many positive integers n |
| N6 |
ROM |
Show that the set of positive integers that cannot be repre- |