Facebook Page
Twitter
RSS
All times are GMT -4. The time now is 23:58.
Results 1 to 5 of 5
  1. MHB Journeyman
    MHB Math Helper
    caffeinemachine's Avatar
    India
    Status
    Offline
    Join Date
    Mar 2012
    Location
    India
    Posts
    561
    Thanks
    450 times
    Thanked
    837 times
    #1
    Each of ten segments has integer length and each one's length is greater than 1cm and less than 55cm. Prove that you can select three sides of a triangle among the segments.

  2. MHB Journeyman
    Inactive Math Helper

    Status
    Offline
    Join Date
    Jan 2012
    Posts
    932
    Thanks
    344 times
    Thanked
    1039 times
    #2
    Quote Originally Posted by caffeinemachine View Post
    Each of ten segments has integer length and each one's length is greater than 1cm and less than 55cm. Prove that you can select three sides of a triangle among the segments.
    Is the wording right? It looks like you can relax the conditions to the lengths being greater than or equal 1cm and less than 55cm.

    CB

  3. MHB Journeyman
    MHB Math Helper
    caffeinemachine's Avatar
    India
    Status
    Offline
    Join Date
    Mar 2012
    Location
    India
    Posts
    561
    Thanks
    450 times
    Thanked
    837 times
    #3
    Quote Originally Posted by CaptainBlack View Post
    Is the wording right? It looks like you can relax the conditions to the lengths being greater than or equal 1cm and less than 55cm.

    CB
    I think you are right. I've taken this from a book and in the book they have the conditions I have posted.

  4. MHB Journeyman
    Inactive Math Helper

    Status
    Offline
    Join Date
    Jan 2012
    Posts
    932
    Thanks
    344 times
    Thanked
    1039 times
    #4
    Quote Originally Posted by caffeinemachine View Post
    I think you are right. I've taken this from a book and in the book they have the conditions I have posted.
    Sorting the lengths into non-decreasing order, and assume that the claim is false, then the ordered sequence of lengths is a super Fibonacci sequence, meaning that:

    \[ l_{k} \ge l_{k-1}+l_{k-2}, k=3, .., 10 \]

    Thus the set of lengths which has the smallest maximum value such that three may not be selected to form a triangle is the Fibonacci sequence, but then \(l_{10}\ge 55\), a contradiction.

    (note I regard the degenerate triangle with two zero angles as a non-triangle, the argument is easily modified if you want this to count as a triangle)

    CB
    Last edited by CaptainBlack; March 23rd, 2012 at 15:57. Reason: fix LaTeX

  5. MHB Journeyman
    MHB Math Helper
    caffeinemachine's Avatar
    India
    Status
    Offline
    Join Date
    Mar 2012
    Location
    India
    Posts
    561
    Thanks
    450 times
    Thanked
    837 times
    #5
    Quote Originally Posted by CaptainBlack View Post
    Sorting the lengths into non-decreasing order, and assume that the claim is false, then the ordered sequence of lengths is a super Fibonacci sequence, meaning that:

    \[ l_{k} \ge l_{k-1}+l_{k-2}, k=3, .., 10 \]

    Thus the set of lengths which has the smallest maximum value such that three may not be selected to form a triangle is the Fibonacci sequence, but then \(l_10\ge 55\), a contradiction.

    (note I regard the degenerate triangle with two zero angles as a non-triangle, the argument is easily modified if you want this to count as a triangle)

    CB
    Nice.

Similar Threads

  1. The set of piecewise continuous functions form a Banach space?
    By bkarpuz in forum Topology and Advanced Geometry
    Replies: 6
    Last Post: September 17th, 2013, 08:06
  2. trig108's Question form Math Help Forum
    By Sudharaka in forum Questions from Other Sites
    Replies: 0
    Last Post: June 16th, 2012, 22:27
  3. fixed point. continuos function from a triangle to a triangle.
    By caffeinemachine in forum Topology and Advanced Geometry
    Replies: 1
    Last Post: May 2nd, 2012, 14:24
  4. Find principal value in a+ib form.
    By GaryBenton in forum Analysis
    Replies: 6
    Last Post: February 8th, 2012, 18:58
  5. No. of ways to form a number
    By Punch in forum Discrete Mathematics, Set Theory, and Logic
    Replies: 4
    Last Post: February 4th, 2012, 09:04

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •  
Math Help Boards