1
If x + y = 1 and x 2 + y 2 = 221 , determine the values of x 3 + y 3 .
2
Given the integers in the 3 X 3 array :
Note that there are 12 adjacent-pair sums, 6 horizontal and 6 vertical. For example, the third row has sums formed from 4 + 6 = 10 and 6 + 8 = 14, and the second column also has 2 adjacent-pair sums formed from 3 + 9 = 12 and 9 + 6 = 15.
(a) Find an arrangement of the numbers in the array so that all 12 adjacent-pair sums are odd.
(b) Show that no possible arrangement of the numbers in the array will make all 12 adjacent-pair sums prime.
3
Determine values for the constants a , b and c where
f (x ) = ax 3 + bx 2 + cx + 9 is a function which is true for all values of x and which satisfies the condition
f (x ) + x 2 = f (x + 1).
4
An arithmetic sequence is one in which the difference between any two successive terms is constant. Consider five numbers which are in arithmetic sequence with the following properties:
(a) the product of the first, third and fifth term is P and the product of the second and the fourth is Q .
(b) the sum of the five terms equals P /Q .
(c) the middle term of the sequence is five times a perfect square.
Show that the product of all five terms is a perfect square.
5
Two circles intersect in the points A(1,4) and B(2,3), and have the y - axis as a common tangent. Determine the equation of the other common tangent.
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