With the first question, please excuse me if I've misread, but aren't there an infinite amount of positive integers such that for ?
I am writing my own AIME-level problems for mock exams and would like some critique based on the difficulty of the problems (e.g. how long it would take, whether it requires a lot of brute force etc.).
1. Let S be the set of positive integers that cannot be written in the form , where a and b are non-negative integers. Find the sum of the elements of S. Ans = 114
2. Points E and F are on sides AB and BC, respectively, of rectangle ABCD, so that AE:EB = 3:1 and BF:FC = 1:2. Segments DE and AF intersect at point G. Find the ratio [EBFG]:[ABCD] ([x] denotes the area of x). Ans = 11/120
3. Compute . Ans = 100/201
Obviously I am not asking you to solve the problems, but if you think one of my answers is incorrect, or you see a really cheap, one-line solution that takes away the challenge from the problem, please let me know. Thank you so much!