MHB Ten segments. One can form a triangle.

[caffeinemachine]

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.

 

[CaptainBlack]

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

 

[caffeinemachine]

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.

 

[CaptainBlack]

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

 

[caffeinemachine]

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.