Prove that amongst 10 sticks of length 1 to 55 there are 3 that form a triangle

Question

I'm trying to prove, that amongst 10 sticks, which length can vary from 1cm to 55cm, there are 3 (or at least 3), using which one can form a triangle. I feel like I should use Dirichlet's pigeon theorem here, but I'm not shure how to apply it. I know that for a triangle to be formed, sum of lengths of two sides must be greater then length of the third, but I still don't know how to apply it here.

Edit: I misread the excercise. The sticks are longer than 1cm and shorter than 55cm. No other information about the sticks is given.

Answer 1

Suppose they don't. Let that be $1\leq a_1\leq a_2\leq ...\leq a_{10}< 55$

Then by the triangle inequality we have $$1+1\leq a_1+a_2\leq a_3\implies a_3\geq 2$$

Then $$1+2\leq a_2+a_3\leq a_4\implies a_4\geq 3$$

and so on, we get $a_5\geq 5$, $a_6\geq 8$, $a_7\geq 13$, $a_8\geq 21$, $a_9\geq 34$ and $a_{10}\geq 55$. A contradiction.