Results 1 to 14 of 14
Like Tree3Thanks
  • 1 Post By HallsofIvy
  • 1 Post By richard1234
  • 1 Post By richard1234

Math Help - Non Degenerate Triangles

  1. #1
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Non Degenerate Triangles

    @mods,
    This is not geometry. This is algebra.

    Non Degenerate Triangles-czech.png

    Then there exist k indices j such that...
    What does this mean? What are indices?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Non Degenerate Triangles

    Indices = plural of index.

    In this case, an "index" is the set b_j, r_j, w_j. For example, b_1, r_1, w_1 and b_{594}, r_{594}, w_{594} are considered to be indices. The question is essentially asking, what is the largest possible k such that the statement is always true, that you can form k non-degenerate triangles with side lengths b_j, r_j, w_j for some j.
    Last edited by richard1234; June 27th 2012 at 10:35 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,952
    Thanks
    1629

    Re: Non Degenerate Triangles

    More commonly, in the United States, called "subscripts".
    Thanks from cosmonavt
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Non Degenerate Triangles

    Another example of a problem using the word "indices":

    2010 USAMO Problems/Problem 3 - AoPSWiki
    Thanks from cosmonavt
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Re: Non Degenerate Triangles

    Thanks guys! Got it! I am a hobbyist programmer. I knew it has something to do with an "array" or a "set" in mathematics. But thanks for the explanation.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Re: Non Degenerate Triangles

    OK so since this is an IMO problem, the solution needs explanation too! :P

    Let's get started.

    Non Degenerate Triangles-solutionczech.png

    1. Now the first part underlined in red, I have no idea. How come does he assume that k is 1?
    2. The second part in red, why does he need to prove that they are always the sides of a triangle? And why only for index 2009? Why not for others?
    3. The first part in black - I know that we can assume this without losing generality, but again, why only 2009? Why not others? Is that because k is 1?
    4. The second part in black, shows that 2009 is non-degenerate right?
    5. The first part in blue. Is he assigning arbitrary variables to members of the set with index 2009? Why?
    6. To the parts in green, I have no idea.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Non Degenerate Triangles

    Quote Originally Posted by cosmonavt View Post
    6. To the parts in green, I have no idea.

    b_i and r_i, 1 \le i \le 2009, are ordered. Use the previous inequalities that are assumed to be true (without loss of generality).
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Non Degenerate Triangles

    Quote Originally Posted by cosmonavt View Post
    OK so since this is an IMO problem, the solution needs explanation too! :P

    Let's get started.

    Click image for larger version. 

Name:	solutionczech.PNG 
Views:	7 
Size:	146.8 KB 
ID:	24184

    1. Now the first part underlined in red, I have no idea. How come does he assume that k is 1?
    2. The second part in red, why does he need to prove that they are always the sides of a triangle? And why only for index 2009? Why not for others?
    3. The first part in black - I know that we can assume this without losing generality, but again, why only 2009? Why not others? Is that because k is 1?
    4. The second part in black, shows that 2009 is non-degenerate right?
    5. The first part in blue. Is he assigning arbitrary variables to members of the set with index 2009? Why?
    6. To the parts in green, I have no idea.
    Sorry, didn't see 1. through 5.

    1. The author's claiming that the maximal k is 1.
    2. b_{2009}, w_{2009}, and r_{2009} are the longest black, white, red sides, but there is no indication that they are of the same triangle.
    3. No. Once you fix an assumption on the ordering, you shouldn't make any other assumptions. It's a little difficult to explain, but I can show you other USAMO/IMO-type problems where you can only make one "WLOG" assumption.
    4. Yes.
    5. He is basically constructing an arbitrary triangle with w_{2009} = w.
    Thanks from cosmonavt
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Re: Non Degenerate Triangles

    Quote Originally Posted by richard1234 View Post
    1. The author's claiming that the maximal k is 1.
    What is the basis of this assumption?

    2. b_{2009}, w_{2009}, and r_{2009} are the longest black, white, red sides, but there is no indication that they are of the same triangle.
    Got it.
    3. No. Once you fix an assumption on the ordering, you shouldn't make any other assumptions. It's a little difficult to explain, but I can show you other USAMO/IMO-type problems where you can only make one "WLOG" assumption.
    Yes, please!
    5. He is basically constructing an arbitrary triangle with w_{2009} = w.
    Shouldn't it be the other way round? If the arbitrary triangle has the length w, then he should assign the value of w_{2009} to w, meaning
    w = w_{2009}
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Re: Non Degenerate Triangles

    Quote Originally Posted by richard1234 View Post
    Another example of a problem using the word "indices":

    2010 USAMO Problems/Problem 3 - AoPSWiki
    I was meaning to ask...

    Just like in this problem you mentioned, the solution requires first making a random assumption and then going onto proof it (instead of approaching the assumption from a definite pathway.

    He assumes that (2i - 1) * (2i) ≤ (2i -1) + (2i) and then goes on to prove it.

    Do these assumptions come from that so called "mental database" of yours? Or is this just the cherry picked part of the solution (and the real solution process consisted of numerous other assumptions by trial and error)?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Non Degenerate Triangles

    Quote Originally Posted by cosmonavt View Post
    Do these assumptions come from that so called "mental database" of yours? Or is this just the cherry picked part of the solution (and the real solution process consisted of numerous other assumptions by trial and error)?
    It's like writing a persuasive paper (more or less) -- You state your original claim, then you back it up with evidence. In this case, the author states his claim (k=1) and proves it.
    Last edited by richard1234; June 30th 2012 at 08:27 AM.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Re: Non Degenerate Triangles

    Yeah but how exactly do you make a claim in the first place? How do you know it WILL be true? Is that because your mental database contains similar problems?

    And also clear my queries in the post #9
    Last edited by cosmonavt; June 29th 2012 at 11:10 PM.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Non Degenerate Triangles

    If it's an IMO problem, you or the author probably would have spent considerable time on this (an hour, maybe two hours). The author has already proven that k=1 using the techniques he is about to describe. Remember, it's just like writing a persuasive paper -- introduce your original claim, and support it with evidence (in this case, proof).
    Last edited by richard1234; June 30th 2012 at 08:27 AM.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Non Degenerate Triangles

    Quote Originally Posted by cosmonavt View Post
    Yes, please!
    An easy one to start: Find all solutions (x,y,z) in non-negative real numbers that satisfy
    x^3 + y^3 = z
    x^3 + z^3 = y
    y^3 + z^3 = x


    Quote Originally Posted by cosmonavt View Post
    Shouldn't it be the other way round? If the arbitrary triangle has the length w, then he should assign the value of w_{2009} to w, meaning
    w = w_{2009}
    I think that's what he did. Besides, w = w_{2009} and w_{2009} = w are equivalent.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Degenerate case of conic?
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: May 19th 2011, 06:22 AM
  2. Solving Degenerate Quadratics
    Posted in the Algebra Forum
    Replies: 7
    Last Post: January 9th 2011, 03:29 AM
  3. non-degenerate covariance matrix?
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: February 19th 2010, 11:45 PM
  4. what does it mean for a vector to be non-degenerate
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: August 25th 2008, 04:23 AM
  5. degenerate
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: November 15th 2006, 08:21 AM

Search Tags


/mathhelpforum @mathhelpforum