2004 JBMO Problems
Problem 1
Prove that the inequality holds for all real numbers and , not both equal to 0.
Problem 2
Let be an isosceles triangle with , let be the midpoint of its side , and let be the line through perpendicular to . The circle through the points , , and intersects the line at the points and . Find the radius of the circumcircle of the triangle in terms of .
Problem 3
If the positive integers and are such that and are both perfect squares, prove that both and are both divisible with .
Problem 4
Consider a convex polygon having vertices, . We arbitrarily decompose the polygon into triangles having all the vertices among the vertices of the polygon, such that no two of the triangles have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the triangle is also a side of the polygon and in white those triangles that have no sides that are sides of the polygon.
Prove that there are two more black triangles that white ones.
See Also
2004 JBMO (Problems • Resources) | ||
Preceded by 2003 JBMO |
Followed by 2005 JBMO | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |