2001 Pan African MO Problems
Contents
Day 1
Problem 1
Find all positive integers such that: is a positive integer.
Problem 2
Let be a positive integer. A child builds a wall along a line with identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?
Problem 3
Let be an equilateral triangle and let be a point outside this triangle, such that is an isoscele triangle with a right angle at . A grasshopper starts from and turns around the triangle as follows. From the grasshopper jumps to , which is the symmetric point of with respect to . From , the grasshopper jumps to , which is the symmetric point of with respect to . Then the grasshopper jumps to which is the symmetric point of with respect to , and so on. Compare the distance and . .
Day 2
Problem 4
Let be a positive integer, and let be a real number. Consider the equation: How many solutions () does this equation have, such that:
Problem 5
Find the value of the sum: where denotes the greatest integer which does not exceed .
Problem 6
Let be a semicircle with centre and diameter .A circle with centre is drawn, tangent to , and tangent to at . A semicircle is drawn, with centre on , tangent to and to . A circle with centre is drawn, internally tangent to and externally tangent to and . Prove that is a rectangle.
See Also
2001 Pan African MO (Problems) | ||
Preceded by 2000 Pan African MO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2002 Pan African MO |
All Pan African MO Problems and Solutions |