2004 Pan African MO Problems

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Day 1

Problem 1

Do there exist positive integers $m$ and $n$ such that: \[3n^2+3n+7=m^3\]

Solution

Problem 2

Is $4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}}$ an integer?

Solution

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.

Solution

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.

Solution

Problem 5

Each of the digits $1$, $3$, $7$ and $9$ 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 $7$.

Solution

Problem 6

Let $ABCD$ be a cyclic quadrilateral such that $AB$ is a diameter of it's circumcircle. Suppose that $AB$ and $CD$ intersect at $I$, $AD$ and $BC$ at $J$, $AC$ and $BD$ at $K$, and let $N$ be a point on $AB$. Show that $IK$ is perpendicular to $JN$ if and only if $N$ is the midpoint of $AB$.

Solution

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
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