2000 Pan African MO Problems
Contents
[hide]Day 1
Problem 1
Solve for :
Problem 2
Define the polynomials by:
Find the coefficient of
in
.
Problem 3
Let and
be coprime positive integers such that:
Prove
is divisible by 2003.
Day 2
Problem 4
Let ,
and
be real numbers such that
, solve the system:
in real numbers
and
.
Problem 5
Let be circle and let
be a point outside
. Let
and
be the tangents from
to
(where
). A line passing through
intersects
at points
and
. Let
be a point on
such that
. Prove that
bisects
.
Problem 6
A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks are available. Find all positive integers such that it is possible to give each of the directors a set of keys to
different locks, according to the requirements and regulations of the company.
See Also
2000 Pan African MO (Problems) | ||
Preceded by First Pan African MO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2001 Pan African MO |
All Pan African MO Problems and Solutions |