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Math Help - Help forming a proof

  1. #1
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    Help forming a proof

    There are nine sticks of different integer lengths,
    each shorter than 55 units. Prove that it is possible to form a triangle
    with three of them.
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  2. #2
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    Quote Originally Posted by Statsnoob2718 View Post
    There are nine sticks of different integer lengths,
    each shorter than 55 units. Prove that it is possible to form a triangle
    with three of them.
    This may be of some help.

    http://www.mathhelpforum.com/math-he...obability.html
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  3. #3
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    Hello, Statsnoob2718!

    There are nine sticks of different integer lengths, each shorter than 55 units.
    Prove that it is possible to form a triangle with three of them.

    Consider a set of 3 sides which does not satisfy the Triangle Inequality.

    Let two sides be (1,2) . . . The third side can be 3 or more.
    . . The 1st non-triangle is (1,2,3).

    Let two sides be (2,3) . . . The third side can be 5 or more.
    . . The 2nd non-triangle is (2,3,5).

    Let two sides be (3,5) . . . The third side can be 8 or more.
    . . The 3rd non-triangle is (3,5,8).

    Let two sides be (5,8) . . . The third side can be 13 or more.
    . . The 4th non-triangle is (5,8,13).


    We see that the non-triangles are three consecutive terms
    . . of the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, . . .


    To have 9 sticks which form non-triangles, their lengths must be
    . . (at least) the first 9 terms of the Fibonacci Sequence:
    . . . . . . 1, 2, 3, 5, 8, 13, 21, 34, 55.


    Since the lengths are < 55, it is possible to form a triangle with three of them.

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