There are two parts to this regular feature: Normal and Abnormal. Normal is open to all school students, while Abnormal is open to everyone, teachers included. Send us your answers with working. The best entries will be mentioned in the next issue. Prizes are $75 and $50 for the Normal section and $100 and $75 for the Abnormal. You are encouraged to send in your solutions even if you cannot answer all questions. The deadline is 31st December 2000.
(Please send in your solutions to different questions on different sheets of paper, with name and address on each.)
Normal Section
1. If a and b are positive numbers, show that [(a+b)/(a2+b2)] ³ [(a2+b2)/(a3+b3)].
2. The straight line segment connecting the mid-points of the opposite sides AB and CD of the convex quadrilateral ABCD is divided into equal half-segments by the diagonal AC. Show that the areas of the triangles ABC and ACD are equal.
3. Are there nineteen consecutive natural numbers whose sum is divisible by 87?
Abnormal Section
1. Find all the natural numbers n such that the number n3+7 is a multiple of n-2.
2. Let x1 and x2 be the roots of the equation 2x2+4ax-a+1 = 0.
(a) Find the maximum value of f(a) = x12+x22 on the interval [-3,-2].
(b) Sketch the graph of f(a).
3. Solve the equation sin2x-2cos2x cosx = 4cosx.