2003 JBMO Problems
Problem 1
Let be a positive integer. A number consists of digits, each of which is 4; and a number consists of digits, each of which is 8. Prove that is a perfect square.
Problem 2
Suppose there are points in a plane no three of which are collinear with the property that if we label these points as in any way whatsoever, the broken line does not intersect itself. Find the maximum value of .
Problem 3
Let , , be the midpoints of the arcs , , on the circumcircle of a triangle not containing the points , , , respectively. Let the line meets and at and , and let be the midpoint of the segment . Let the line meet and at and , and let be the midpoint of the segment .
a) Find the angles of triangle ;
b) Prove that if is the point of intersection of the lines and , then the circumcenter of triangle lies on the circumcircle of triangle .
Problem 4
Let . Prove that
See Also
2003 JBMO (Problems • Resources) | ||
Preceded by 2002 JBMO |
Followed by 2004 JBMO | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |