Mathematical miniatures

Andreescu, Titu; Savchev, Svetoslav9780883856000, 088385600X, 9780883856451, 088385645X

This is a mathematics book, not a programming book, although it explains Pascal to beginners. It is aimed at high school students and undergraduates with a strong interest in mathematics, and teachers looking for fresh ideas. It is full of diverse mathematical ideas requiring little background. It includes a large number of challenging problems, many of which illustrate how numerical computation leads to conjectures which can then be proved by mathematical reasoning. It is assumed that readers have a PC at their disposal.

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Table of contents :
1. A telescoping sum —
2. Lagrange’s identity —
3. Perfect squares —
4. Lest common multiples —
5. Trig substitutions —
Coffee break 1 —
6. Popoviciu’s theorem —
7. Catalan’s identity —
8. Several inequalities —
9. Vectors —
10. Mathematical induction at work —
Coffee break 2 —
11. A highly divisible determinant —
12. Hermite’s identity —
13. Complete sequences —
14. Three polynomials —
15. More about induction —
Coffee break 3 —
16. A classical identity —
17. Multiplicative functions —
18. The “arbitrary” Proizvolov —
19. Hölder’s inequality —
20. Symmetry —
Coffee break 4 —
21. He knows I know he knows —
22. A special inequality —
23. Two inductive constructions —
24. Some old-fashioned geometry —
25. Extremal arguments —
Coffee break 5 —
26. The AMS inequality —
27. Helly’s theorem for one dimension —
28. Two approaches —
29. Radical axis —
30. The pigeonhole principle —
Coffee break 6 —
31. The three jug problem —
32. Rectifying trajectories —
33. Numerical systems —
34. More on polynomials —
35. Geometric transformations —
Coffee break 7 —
36. The Game of life problem —
37. Tetrahedra with a point in common —
38. Should we count —
39. Let’s count now —
40. Some elementary number theory —
Coffee break 8 —
41. Euclid’s game —
42. Perfect powers —
43. The 2n-1 problem —
44. The 2n+1 problem —
45. The 3n problem —
Coffee break —
46. Pairwise sums —
47. Integer progressions —
48. Incomparable sets —
49. Morse’s sequence —
50. A favorite of Erdös.