IMO 2001 Shortlist
28 problems · Source: IMO Compendium
Algebra
| # |
Origin |
Problem |
| A1 |
IND |
Let T denote the set of all ordered triples (p, q, r) of nonneg- |
| A2 |
POL |
Let a0, a1, a2, . . . be an arbitrary infinite sequence of positive |
| A3 |
ROM |
Let x1, x2, . . . , xn be arbitrary real numbers. Prove the |
| A4 |
LIT |
Find all functions f : R \toR satisfying |
| A5 |
BUL |
Find all positive integers a1, a2, . . . , an such that |
| A6 |
KOR |
Prove that for all positive real numbers a, b, c, |
Combinatorics
| # |
Origin |
Problem |
| C1 |
COL |
Let A = (a1, a2, . . . , a2001) be a sequence of positive integers. |
| C2 |
CAN |
Let n be an odd integer greater than 1 and let c1, c2, . . . , |
| C3 |
RUS |
Define a k-clique to be a set of k people such that every pair |
| C4 |
NZL |
A set of three nonnegative integers {x, y, z} with x < y < z |
| C5 |
FIN |
Find all finite sequences (x0, x1, . . . , xn) such that for every |
| C6 |
CAN |
For a positive integer n define a sequence of zeros and ones |
| C7 |
FRA |
A pile of n pebbles is placed in a vertical column. This |
| C8 |
GER |
Twenty-one girls and twenty-one boys took part in a |
Geometry
| # |
Origin |
Problem |
| G1 |
UKR |
Let A1 be the center of the square inscribed in acute triangle |
| G2 |
KOR |
In acute triangle ABC with circumcenter O and altitude |
| G3 |
GBR |
Let ABC be a triangle with centroid G. Determine, with |
| G4 |
FRA |
Let M be a point in the interior of triangle ABC. Let A′ lie |
| G5 |
GRE |
Let ABC be an acute triangle. Let DAC, EAB, and FBC |
| G6 |
IND |
Let ABC be a triangle and P an exterior point in the plane |
| G7 |
BUL |
Let O be an interior point of acute triangle ABC. Let A1 |
| G8 |
ISR |
Let ABC be a triangle with ∡BAC = 60◦. Let AP bisect |
Number Theory
| # |
Origin |
Problem |
| N1 |
AUS |
Prove that there is no positive integer n such that for k = |
| N2 |
COL |
Consider the system |
| N3 |
GBR |
Let a1 = 1111, a2 = 1212, a3 = 1313, and |
| N4 |
VIE |
Let p \geq5 be a prime number. Prove that there exists an |
| N5 |
BUL |
Let a > b > c > d be positive integers and suppose |
| N6 |
RUS |
Is it possible to find 100 positive integers not exceeding |