IMO 2003 Shortlist
27 problems · Source: IMO Compendium
Algebra
| # |
Origin |
Problem |
| A1 |
USA |
Let aij, i = 1, 2, 3, j = 1, 2, 3, be real numbers such that aij |
| A2 |
AUS |
Find all nondecreasing functions f : R \toR such that |
| A3 |
GEO |
Consider pairs of sequences of positive real numbers a1 \geq |
| A4 |
IRE |
Let n be a positive integer and let x1 \leqx2 \leq\cdot \cdot \cdot \leqxn be |
| A5 |
KOR |
Let R+ be the set of all positive real numbers. Find all |
| A6 |
USA |
Let n be a positive integer and let (x1, . . . , xn), (y1, . . . , yn) |
Combinatorics
| # |
Origin |
Problem |
| C1 |
BRA |
Let A be a 101-element subset of the set S = {1, 2, . . . , |
| C2 |
GEO |
Let D1, . . . , Dn be closed disks in the plane. (A closed disk |
| C3 |
LIT |
Let n \geq5 be a given integer. Determine the largest integer |
| C4 |
IRN |
Let x1, . . . , xn and y1, . . . , yn be real numbers. Let A = |
| C5 |
ROM |
Every point with integer coordinates in the plane is the |
| C6 |
SAF |
Let f(k) be the number of integers n that satisfy the following |
Geometry
| # |
Origin |
Problem |
| G1 |
FIN |
Let ABCD be a cyclic quadrilateral. Let P, Q, R be the |
| G2 |
GRE |
Three distinct points A, B, C are fixed on a line in this order. |
| G3 |
IND |
Let ABC be a triangle and let P be a point in its interior. |
| G4 |
ARM |
Let \Gamma1, \Gamma2, \Gamma3, \Gamma4 be distinct circles such that \Gamma1,… |
| G5 |
KOR |
Let ABC be an isosceles triangle with AC = BC, whose |
| G6 |
POL |
Each pair of opposite sides of a convex hexagon has the |
| G7 |
SAF |
Let ABC be a triangle with semiperimeter s and inradius |
Number Theory
| # |
Origin |
Problem |
| N1 |
POL |
Let m be a fixed integer greater than 1. The sequence |
| N2 |
USA |
Each positive integer a undergoes the following procedure in |
| N3 |
BUL |
Determine all pairs (a, b) of positive integers such that |
| N4 |
ROM |
Let b be an integer greater than 5. For each positive integer |
| N5 |
KOR |
An integer n is said to be good if |n| is not the square of |
| N6 |
FRA |
Let p be a prime number. Prove that there exists a prime |
| N7 |
BRA |
The sequence a0, a1, a2, . . . is defined as follows: |
| N8 |
IRN |
Let p be a prime number and let A be a set of positive integers |