IMO 2002 Shortlist
27 problems · Source: IMO Compendium
Algebra
| # |
Origin |
Problem |
| A1 |
CZE |
Find all functions f from the reals to the reals such that |
| A2 |
YUG |
Let a1, a2, . . . be an infinite sequence of real numbers for |
| A3 |
POL |
Let P be a cubic polynomial given by P(x) = ax3+bx2+cx+ |
| A4 |
IND |
Find all functions f from the reals to the reals such that |
| A5 |
IND |
Let n be a positive integer that is not a perfect cube. Define |
| A6 |
IRN |
Let A be a nonempty set of positive integers. Suppose that |
Combinatorics
| # |
Origin |
Problem |
| C1 |
COL |
Let n be a positive integer. Each point (x, y) in the plane, |
| C2 |
ARM |
For n an odd positive integer, the unit squares of an n \times n |
| C3 |
COL |
Let n be a positive integer. A sequence of n positive integers |
| C4 |
BUL |
Let T be the set of ordered triples (x, y, z), where x, y, z are |
| C5 |
BRA |
Let r \geq2 be a fixed positive integer, and let F be an infinite |
| C6 |
POL |
Let n be an even positive integer. Show that there is a |
| C7 |
NZL |
Among a group of 120 people, some pairs are friends. A weak |
Geometry
| # |
Origin |
Problem |
| G1 |
FRA |
Let B be a point on a circle S1, and let A be a point distinct |
| G2 |
KOR |
Let ABC be a triangle for which there exists an interior |
| G3 |
KOR |
The circle S has center O, and BC is a diameter of S. |
| G4 |
RUS |
Circles S1 and S2 intersect at points P and Q. Distinct points |
| G5 |
AUS |
For any set S of five points in the plane, no three of which |
| G6 |
UKR |
Let n \geq3 be a positive integer. Let C1, C2, C3, . . . , Cn |
| G7 |
BUL |
The incircle Ωof the acute-angled triangle ABC is tangent |
| G8 |
ARM |
Let S1 and S2 be circles meeting at the points A and B. A |
Number Theory
| # |
Origin |
Problem |
| N1 |
UZB |
What is the smallest positive integer t such that there exist |
| N2 |
ROM |
Let n \geq2 be a positive integer, with divisors 1 = d1 < |
| N3 |
MON |
Let p1, p2, . . . , pn be distinct primes greater than 3. Show |
| N4 |
GER |
Is there a positive integer m such that the equation |
| N5 |
IRN |
Let m, n \geq2 be positive integers, and let a1, a2, . . . , an |
| N6 |
ROM |
Find all pairs of positive integers m, n \geq3 for which |