IMO 1996 Shortlist
30 problems · Source: IMO Compendium
Algebra
| # |
Origin |
Problem |
| A1 |
SLO |
Let …, …, and … be positive real numbers such that …. |
| A2 |
IRE |
Let … be real numbers such that |
| A3 |
GRE |
Let … be given, and define recursively |
| A4 |
KOR |
Let … be nonnegative real numbers, not all zero. |
| A5 |
ROM |
Let … be the real polynomial function |
| A6 |
IRE |
Let … be an even positive integer. Prove that there exists a positive integer …… |
| A7 |
ARM |
Let … be a function from the set of real numbers … into itself such that for… |
| A8 |
ROM |
Let … denote the set of nonnegative integers. Find all functions … such that |
| A9 |
POL |
Let the sequence …, …, be generated as follows: |
Combinatorics
| # |
Origin |
Problem |
| C1 |
FIN |
We are given a positive integer … and a rectangular board … with dimensions …,… |
| C2 |
UKR |
An … square is divided into … unit squares in the usual manner. Each of the …… |
| C3 |
USA |
Let … be integers such that …. Determine the maximum size of a subset … of the… |
| C4 |
FIN |
Determine whether or not there exist two disjoint infinite sets … and … of… |
| C5 |
FRA |
Let … be three positive integers with …. |
| C6 |
CAN |
A finite number of beans are placed on an infinite row of squares. A sequence… |
| C7 |
IRE |
Let … be a finite set and let …, … be bijective functions from … onto itself.… |
Geometry
| # |
Origin |
Problem |
| G1 |
GBR |
Let … have orthocenter …, and let … be a point on its circumcircle, distinct… |
| G2 |
CAN |
Let … be a point inside … such that |
| G3 |
GBR |
Let … be an acute-angled triangle with …. Let … be the circumcenter, … its… |
| G4 |
USA |
Let … be an equilateral triangle and let … be a point in its interior. Let the… |
| G5 |
ARM |
Let … be a convex hexagon such that …, …, and …. Let …, …, … be the circumradii… |
| G6 |
ARM |
Let the sides of two rectangles be … and … with |
| G7 |
GBR |
Let … be an acute-angled triangle with circumcenter … and circumradius …. Let …… |
| G8 |
RUS |
Let … be a convex quadrilateral, and let …, …, …, and … denote the circumradii… |
| G9 |
UKR |
In the plane are given a point … and a polygon … (not necessarily convex). Let… |
Number Theory
| # |
Origin |
Problem |
| N1 |
UKR |
Four integers are marked on a circle. At each step we simultaneously replace… |
| N2 |
RUS |
The positive integers … and … are such that the numbers |
| N3 |
BUL |
A finite sequence of integers … is called quadratic if for each … we have the… |
| N4 |
BUL |
Find all positive integers … and … for which |
| N5 |
ROM |
Let … denote the set of nonnegative integers. Find a bijective function … from… |