Mock AIME questions -- critique

Hi everyone,

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!

richard1234

With the first question, please excuse me if I've misread, but aren't there an infinite amount of positive integers such that for ?

5a+7b. And there are only a finite number of positive integers that cannot be written in this form.

But you say a and b are any non-negative integers. I can choose an infinite number of a and b!

Richard meant that there are only a finite number of positive integers that can NOT be written as 5a+7b. See this thread.

Yes, basically what emakarov said. However for this case you don't need numerical semi-groups (although if you know them, more power to you).