# Thread: Help forming a proof

1. ## 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.

2. Originally Posted by 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.
This may be of some help.

http://www.mathhelpforum.com/math-he...obability.html

3. 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.