2005 JBMO Problems
Problem 1
Find all positive integers satisfying the equation
Problem 2
Let be an acute-angled triangle inscribed in a circle . It is given that the tangent from to the circle meets the line at point . Let be the midpoint of the line segment and be the second intersection point of the circle with the line . The line meets again the circle at point different from .
Prove that the lines and are parallel.
Problem 3
Prove that there exist
(a) 5 points in the plane so that among all the triangles with vertices among these points there are 8 right-angled ones;
(b) 64 points in the plane so that among all the triangles with vertices among these points there are at least 2005 right-angled ones.
Problem 4
Find all 3-digit positive integers such that where is the decimal representation of the number.
See Also
2005 JBMO (Problems • Resources) | ||
Preceded by 2004 JBMO |
Followed by 2006 JBMO | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |