2004 Pan African MO Problems
Contents
[hide]Day 1
Problem 1
Do there exist positive integers and
such that:
Problem 2
Is an integer?
Problem 3
One writes 268 numbers around a circle, such that the sum of 20 consecutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.
Day 2
Problem 4
Three real numbers satisfy the following statements:
(1) the square of their sum equals to the sum their squares.
(2) the product of the first two numbers is equal to the square of the third number.
Find these numbers.
Problem 5
Each of the digits ,
,
and
occurs at least once in the decimal representation of some positive integers. Prove that one can permute the digits of this integer such that the resulting integer is divisible by
.
Problem 6
Let be a cyclic quadrilateral such that
is a diameter of it's circumcircle. Suppose that
and
intersect at
,
and
at
,
and
at
, and let
be a point on
. Show that
is perpendicular to
if and only if
is the midpoint of
.
See Also
2004 Pan African MO (Problems) | ||
Preceded by 2003 Pan African MO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2005 Pan African MO |
All Pan African MO Problems and Solutions |