Hints for the problems at “Using simple special cases in problem-solving”.
1. Suppose it’s a 1×1 square. Which of the inequalities is valid? And suppose it is a very tiny square (say side 0.0001). Which of the inequalities is valid?
You can also do this by symmetry-type arguments. For any given shape of rectangle, A increases with the square of P. So the “shape” of the answer must be an equation connecting what power of P with what power of A?
Or another symmetry-type argument. The greatest rectangle area for any given perimeter P is got by a square. (Generally, the greatest polygon area for any given perimeter is got by a regular polygon, and the greatest area overall for any given perimeter is got by a circle). So A≤(the area of a square with perimeter P.
2. Think about the maximum area when θ=π/2, and when θ is very small.
3. Think about n=1.
4. Can y be negative? What value are all its maxima? Are its maxima equally spaced?
5. Never mind about 100 for the moment. Find n for when the sum≥2, then n for when the sum≥3, then n for when the sum ≥4. See the pattern…
6. Never mind about 99 and 100 for the present. Compare:
12 and 21
23 and 32
34 and 43
See the trend. If you want to be sure the trend is real, work out:
7. Never mind about 1,000,000 for the moment. Work out the 2-and-5 problem for 0≤n<10, and the 3-and-7 problem for 0≤,n<21. See if you can build on that.