2004 Pan African MO Problems
Do there exist positive integers and such that:
Is an integer?
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.
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.
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 .
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 .
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