tailieunhanh - Number Theory Problems (The J29 Project)

1. Show that there exist innitely many non similar triangles such that the side-lengths are positive integers and the areas of squares constructed on their sides are in arithmetic progression. 2. Let n be a positive integer. Find the number of those numbers of 2n digits in the binary system for which the sum of digits in the odd places is equal to the sum of digits in the even places. | Number Theory Problems The J29 Project Amir Hossein Parvardi July 11 2012 Contents 1 Problems 5 Amir Hossein. 5 Amir Hossein - Part 1. 5 Amir Hossein - Part 2. 7 Amir Hossein - Part 3. 9 Amir Hossein - Part 4. 11 Amir Hossein - Part 5 . 13 Amir Hossein - Part 6. 14 Amir Hossein - Part 7. 16 Amir Hossein - Part 8. 18 Amir Hossein - Part 9. 20 Amir Hossein - Part 10 . 22 Amir Hossein - Part 11 . 24 Amir Hossein - Part 12 . 26 Amir Hossein - Part 13 . 28 Amir Hossein - Part 14 . 30 Amir Hossein - Part 15 . 32 Amir Hossein - Part 16 . 34 Amir Hossein - Part 17 . 35 Amir Hossein - Part 18 . 37 Amir Hossein - Part 19 . 39 Andrew. 40 Andrew - Part 1 . 40 Andrew - Part 2 . 42 Andrew - Part 3 . 44 Andrew - Part 4 . 46 Andrew - Part 5 . 47 Andrew - Part 6 . 49 Andrew - Part 7 . 51 Andrew - Part 8 . 53 Andrew - Part 9 . 55 Goutham . 56 Goutham - Part 1 . 56 Goutham - Part 2 . 58 1 2 Goutham - Part 3 . 61 Goutham - Part 4 . 62 Goutham - Part 5 . 64 Goutham - Part 6 . 66 Orlando. 67 Orlando - Part 1. 67 Orlando - Part 2. 69 Orlando - Part 3 . 71 Orlando - Part 4 . 73 Orlando - Part 5 . 75 Orlando - Part 6 . 77 Orlando - Part 7 . 79 Orlando - Part 8. 80 Orlando - Part 9. 82 Orlando - Part 10 . 83 Valentin. 85 Valentin - Part 1. 85 Valentin - Part 2 . 87 Valentin - Part 3 . 89 Valentin - Part 4 . 91 Darij. 93 Darij - Part 1. 93 Darij - Part 2 . 95 Vesselin . 98 Vesselin - Part 1 . 98 Gabriel . 99 Gabriel - Part 1 . 99 Gabriel - Part 2 . 101 April - Part April - Part 2 . 104 April - Part 3 . 106 Arne . 108 Arne - Part 1 . 108 Arne - Part 2 . 110 Kunihiko .111