| 21. Let n be a positive integer. Prove that if m = 0 (mod 3), then m = 2r (mod 3n) for some positive integer r. | |
| 22. How many square residues are there mod 2n? | |
| 23. What is the maximum value of 1/2x + 1/21/x for x > 0? | |
| 24. Prove that n distinct points in the plane, but not all on a single line, determine at least n distinct lines. | |
| 25. [0, 1] is the closed real interval. f:[0, 1] → [0, 1] is monotonic increasing, and f(0) = 0, f(1) = 1. Prove that the graph of f can be covered by n rectangles with sides parallel to the x-axis and y-axis and with area 1/n2. | |
| 26. Prove that any finite set of closed squares with total area 3 can be arranged to cover the unit square. | |
| 27. Prove that any finite set of closed squares with total area 1/2 can be fitted inside a unit square without overlapping. | |
| 28. Find the smallest subset X of the plane such that no point of the plane is at a rational distance from all points of X. | |
| 29. x1, x2, x3, ... are distinct points in the plane such that the distance between any two points is an integer. Prove that all the points are collinear. | |
| 30. A and B are positive integers. xi = (A + 1/2)i + (B + 1/2)i. Prove that only finitely many of x1, x2, x3, ... are integers. |
Donald J Newman, A problem seminar (Springer, problem books in mathematics, 1982). Highly recommended. It comes with short hints, giving the key idea and solutions which are concise but motivated (in other words, they often explain how you might have found the solution yourself).
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John Scholes
jscholes@kalva.demon.co.uk
25 Sep 1999