2003 IMO Problems
Problems of the 2003 IMO.
Contents
Day I
Problem 1
is the set . Show that for any subset of with elements we can find distinct elements of , such that the sets are all pairwise disjoint.
Problem 2
Determine all pairs of positive integers such that is a positive integer.
Problem 3
Each pair of opposite sides of convex hexagon has the property that the distance between their midpoints is times the sum of their lengths. Prove that the hexagon is equiangular.
Day II
Problem 4
Let be a cyclic quadrilateral. Let , , and be the feet of perpendiculars from to lines , , and , respectively. Show that if and only if the bisectors of angles and meet on segment .
Problem 5
Let be a positive integer and let be real numbers. Prove that
with equality if and only if form an arithmetic sequence.
Problem 6
Let be a prime number. Prove that there exists a prime number such that for every integer , the number is not divisible by .
See Also
2003 IMO (Problems) • Resources | ||
Preceded by 2002 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2004 IMO |
All IMO Problems and Solutions |