IMO 1994 Shortlist
24 problems · Source: IMO Compendium
Algebra
| # |
Origin |
Problem |
| A1 |
USA |
Let a0 = 1994 and an+1 = |
| A2 |
FRA |
Let m and n be positive integers. The set A = {a1, a2, . . . , |
| A3 |
GBR |
Let S be the set of real numbers greater than −1. Find |
| A4 |
MON |
Let R denote the set of all real numbers and R+ the subset |
| A5 |
POL |
Let f(x) = x2+1 |
Combinatorics
| # |
Origin |
Problem |
| C1 |
UKR |
On a 5 \times 5 board, two players alternately mark numbers on |
| C2 |
COL |
In a certain city, age is reckoned in terms of real numbers |
| C3 |
MCD |
Peter has three accounts in a bank, each with an integral |
| C4 |
EST |
There are n + 1 fixed positions in a row, labeled 0 to n in |
| C5 |
SWE |
At a round table are 1994 girls, playing a game with a deck |
| C6 |
FIN |
On an infinite square grid, two players alternately mark sym- |
| C7 |
BRA |
Prove that for any integer n \geq2, there exists a set of 2n−1 |
Geometry
| # |
Origin |
Problem |
| G1 |
FRA |
A semicircle \Gamma is drawn on one side of a straight line l. C |
| G2 |
UKR |
ABCD is a quadrilateral with BC parallel to AD. M is the |
| G3 |
RUS |
A circle \omega is tangent to two parallel lines l1 and l2. A second |
| G4 |
AUS-ARM |
N is an arbitrary point on the bisector of \angleBAC. |
| G5 |
CYP |
A line l does not meet a circle \omega with center O. E is the |
Number Theory
| # |
Origin |
Problem |
| N1 |
BUL |
M is a subset of {1, 2, 3, . . ., 15} such that the product of |
| N2 |
AUS |
Determine all pairs (m, n) of positive integers such that |
| N3 |
FIN |
Find a set A of positive integers such that for any infinite |
| N4 |
FRA |
For any positive integer x0, three sequences {xn}, {yn}, and |
| N5 |
ROM |
For any positive integer k, Ak is the subset of {k+1, k+ |
| N6 |
LAT |
Let x1 and x2 be relatively prime positive integers. For n \geq2, |
| N7 |
GBR |
A wobbly number is a positive integer whose digits in base 10 |