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 $\displaystyle 5a+7b$, 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 $\displaystyle \sum_{k=1}^{100} \frac{1}{4k^2 - 1}$. 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

Re: Mock AIME questions -- critique

With the first question, please excuse me if I've misread, but aren't there an infinite amount of positive integers $\displaystyle n$ such that $\displaystyle n \neq 5a + 7b$ for $\displaystyle a, b \in \mathbb{N}$?

Re: Mock AIME questions -- critique

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

Re: Mock AIME questions -- critique

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

Re: Mock AIME questions -- critique

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

Re: Mock AIME questions -- critique

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).