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Thread: Extracurricular Elementary Mathematics Marathon

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    -insert title here- Paradoxica's Avatar
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    Extracurricular Elementary Mathematics Marathon

    Rules are as per the other marathons. Difficulty should be reasonable, and hints should be provided as deemed necessary. Use any "elementary" techniques, provided you state them, and if it's fairly advanced, outline a proof.

    I'll start off simple.



    Other things:

    For the purposes of this thread, the set of all natural numbers excludes zero.

    Vectors and elementary functions of any kind are allowed, but little to no calculus. (this one due to leehuan)

    Define terms that the average person following this thread probably wouldn't know.

    State any theorems/techniques that may help in solving the problem.
    Last edited by Paradoxica; 22 Jan 2016 at 9:03 PM.
    leehuan likes this.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    Rules are as per the other marathons. Difficulty should be reasonable, and hints should be provided as deemed necessary. Use any "elementary" techniques, provided you state them, and if it's fairly advanced, outline a proof.

    I'll start off simple.





















    VBN2470 likes this.

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    Re: Extracurricular Elementary Mathematics Marathon




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    Señor Member GoldyOrNugget's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by InteGrand View Post


    16

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by GoldyOrNugget View Post
    16
    Correct!

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    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by InteGrand View Post


    The answer, has already been posted, but here, we will prove uniqueness.

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon



    Hint: Use Fermat's Little Theorem.
    Last edited by Paradoxica; 21 Jan 2016 at 10:24 AM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    The answer, has already been posted, but here, we will prove uniqueness.

    VBN2470 likes this.

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by InteGrand View Post
    That doesn't prove uniqueness of b, it just restricts the value range of b to the solution interval without proof...
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Loquacious One Drsoccerball's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Isn't this just the Advanced X2 Marathon ?

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    That doesn't prove uniqueness of b, it just restricts the value range of b to the solution interval without proof...
    VBN2470 likes this.

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Drsoccerball View Post
    Isn't this just the Advanced X2 Marathon ?
    No, it is unrestricted from the arbitrary bounds placed on it. I can't just post any Olympiad problem on there, but I can do so here.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Drsoccerball View Post
    Isn't this just the Advanced X2 Marathon ?
    If this were my thread I would allow hyperbolic functions and elementary vector notation now, however probably a bit of guidance as to how they work as we haven't attended a lecture yet.
    Last edited by leehuan; 21 Jan 2016 at 10:40 AM.

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    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by leehuan View Post
    If this were my thread I would allow hyperbolic functions and elementary vector notation now, however probably a bit of guidance as to how they work as we haven't attended a lecture yet.
    Olympiad allows for that, but I've never seen it in good use. Except for vectors, those things are really useful for a good bunch of problems.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post


    Hint: Use Fermat's Little Theorem.
    p-1 = mn + r for non-negative integers m, r with r < n.

    Then (a^r)(a^n)^m=a^(p-1)
    => a^r=1 (using FLT)

    From the minimality of n, we must conclude r=0.

    That is n|p-1.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Solve the equation x^2+y^2+1=xyz over the positive integers.

    Hint: First concentrate on determining what possibilities there are for z.
    Last edited by seanieg89; 22 Jan 2016 at 8:11 AM.

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by seanieg89 View Post
    Solve the equation x^2+y^2=xyz over the positive integers.

    Hint: First concentrate on determining what possibilities there are for z.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  18. #18
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  19. #19
    Señor Member GoldyOrNugget's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    Trivial for n=1.

    Suppose the property holds for 1..n-1.

    For general n, if their sum is divisible by n, then their sum mod n is 0. Take all elements mod n. If all elements mod n are 0, pick any subset. Otherwise, pick out an element x mod N which is nonzero mod n. 0 <= x < n. Of the remaining elements, take any n-x > 0 elements. These n-x elements have a subset divisible by n-x, so if we then add x to the subset, we have a subset with sum (n-x) + x = 0 mod N, so the subset is divisible by n.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    Nice . I actually forgot a term on the LHS from the diophantine equation I was trying to remember though lol, try the edited problem too.

    It is slightly harder, but not greatly so. The original hint still stands.

    Repost for visibility:

    Solve the equation:



    over the positive integers.

    Hint: Try to determine what z can be first.
    Last edited by seanieg89; 22 Jan 2016 at 8:18 AM.

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    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by seanieg89 View Post
    Nice . I actually forgot a term on the LHS from the diophantine equation I was trying to remember though lol, try the edited problem too.

    It is slightly harder, but not greatly so. The original hint still stands.

    Repost for visibility:

    Solve the equation:



    over the positive integers.

    Hint: Try to determine what z can be first.




    Last edited by Paradoxica; 22 Jan 2016 at 9:01 PM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post




    Well done .

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by GoldyOrNugget View Post
    Trivial for n=1.

    Suppose the property holds for 1..n-1.

    For general n, if their sum is divisible by n, then their sum mod n is 0. Take all elements mod n. If all elements mod n are 0, pick any subset. Otherwise, pick out an element x mod N which is nonzero mod n. 0 <= x < n. Of the remaining elements, take any n-x > 0 elements. These n-x elements have a subset divisible by n-x, so if we then add x to the subset, we have a subset with sum (n-x) + x = 0 mod N, so the subset is divisible by n.
    I don't think this is quite right? Just because the sum of a subset is divisible by n-x, it doesn't necessarily mean that the sum is n-x mod N? Or have I misinterpreted what you're saying?

    Here's what I had. Suppose that a_i are the elements of the set. Now, consider the sums b_i=a_1+a_2+....+a_i. If one of b_1,b_2.. b_n is 0 mod n, that set has sum divisible by n. If not, then by the pigeon hole principle, two of these are of the same class mod n. Suppose that these two are b_m and b_n, (n>m). Then, b_n-b_m has 0 mod n.

    Quote Originally Posted by Paradoxica View Post


    I think we need a bit of clarification here, \pi (3)= 2, but we can definitely have subintervals of length 3 which contain no prime numbers. For instance 20,21,22 is such an interval, and none of the three are prime numbers.
    Last edited by Blast1; 26 Jan 2016 at 2:26 PM.

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    Yes, there can be sub-intervals of length "n" with no prime numbers. however, you must prove this for all n.

    And the second part is less straightforward, but still doable. The upper bound is always pi(n), because that is the maximum number of prime numbers in any sub-interval of the natural numbers of length n
    ?

    In my previous post, I was addressing the second part of your question, not the first (I suppose I should've mentioned this). Basically, in your question you claim that if we have a subinterval of length n (has n elements), then there must exist a subinterval containing k primes for all k which exceeds 0 but is bounded by pi(x). In my previous post, I provided a counter example to this claim, showing how there exists a subinterval of length 3, but within the subinterval which I specifically mentioned (20,21,22) there is no subinterval which contains any primes. According to your question, within the subinterval (20,21,22), there should exist a subinterval which contains 1 prime and also another subinterval which contains 2 primes. Obviously, that's not true in this example.


    Also, the first part of your question doesn't quite make sense either. For instance, the elements 2 and 3 form a subinterval of length 2, and clearly any subinterval within this subinterval must contain a prime. Thus your claim is false.

    So this is why I'm asking for a clarification, as the question in its current wording isn't true.
    Last edited by Blast1; 27 Jan 2016 at 1:08 AM.

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    Re: Extracurricular Elementary Mathematics Marathon



    zz
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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