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Math Help - Mock AIME questions -- critique

  1. #1
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    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 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 \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
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  2. #2
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    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 n such that n \neq 5a + 7b for a, b \in \mathbb{N}?
    Last edited by rowe; July 3rd 2012 at 08:12 AM.
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  3. #3
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    Re: Mock AIME questions -- critique

    5a+7b. And there are only a finite number of positive integers that cannot be written in this form.
    Last edited by richard1234; July 3rd 2012 at 08:28 AM.
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  4. #4
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    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!
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    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.
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  6. #6
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    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).
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