2001 Pan African MO Problems
Contents
[hide]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 |