IMO 2004 Shortlist
30 problems · Source: IMO Compendium
Algebra
| # |
Origin |
Problem |
| A1 |
KOR |
Let n \geq3 be an integer and t1, t2, . . . , tn positive real |
| A2 |
ROM |
An infinite sequence a0, a1, a2, . . . of real numbers satisfies |
| A3 |
CAN |
Does there exist a function s: Q … such that if x |
| A4 |
KOR |
Find all polynomials P(x) with real coefficients that |
| A5 |
THA |
Let a, b, c > 0 and ab + bc + ca = 1. Prove the inequality |
| A6 |
RUS |
Find all functions f : R \toR satisfying the equation |
| A7 |
IRE |
Let a1, a2, . . . , an be positive real numbers, n > 1. Denote by |
Combinatorics
| # |
Origin |
Problem |
| C1 |
PUR |
There are 10001 students at a university. Some students join |
| C2 |
GER |
Let n and k be positive integers. There are given n circles |
| C3 |
AUS |
The following operation is allowed on a finite graph: Choose |
| C4 |
POL |
Consider a matrix of size n\timesn whose entries are real numbers |
| C5 |
NZL |
Let N be a positive integer. Two players A and B, taking |
| C6 |
IRN |
For an n \times n matrix A, let Xi be the set of entries in row |
| C7 |
EST |
Determine all m \times n rectangles that can be covered with |
| C8 |
POL |
For a finite graph G, let f(G) be the number of triangles |
Geometry
| # |
Origin |
Problem |
| G1 |
ROM |
Let ABC be an acute-angled triangle with AB ̸= AC. |
| G2 |
KAZ |
The circle \Gamma and the line ℓdo not intersect. Let AB be the |
| G3 |
KOR |
Let O be the circumcenter of an acute-angled triangle ABC |
| G4 |
POL |
In a convex quadrilateral ABCD the diagonal BD does |
| G5 |
SMN |
Let A1A2 . . . An be a regular n-gon. The points B1, . . . , Bn−1 |
| G6 |
GBR |
Let P be a convex polygon. Prove that there is a convex |
| G7 |
RUS |
For a given triangle ABC, let X be a variable point on |
| G8 |
SMN |
A cyclic quadrilateral ABCD is given. The lines AD and |
Number Theory
| # |
Origin |
Problem |
| N1 |
BLR |
Let \tau(n) denote the number of positive divisors of the positive |
| N2 |
RUS |
The function \psi from the set N of positive integers into itself |
| N3 |
IRN |
A function f from the set of positive integers N into itself is |
| N4 |
POL |
Let k be a fixed integer greater than 1, and let m = 4k2 −5. |
| N5 |
IRN |
We call a positive integer alternate if its decimal digits |
| N6 |
IRE |
Given an integer n > 1, denote by Pn the product of all |
| N7 |
BUL |
Let p be an odd prime and n a positive integer. In the |