Given $n$ sticks. From any three, an obtuse triangle can be formed. What is the largest possible value of $n$?
From three sticks of lengths $a \leq b \leq c$, a triangle can be formed if $a+b>c$. According to the cosine theorem, this triangle is obtuse if and only if $a^{2}+b^{2}<c^{2}$. For the given problem, we need to check if $a_{4}^{2}+a_{3}^{2} \geqslant 2 a_{3}^{2}>2 a_{2}^{2}+2 a_{1}^{2}$. On the other hand, $a_{5}<a_{1}+a_{2}$, from which $a_{5}^{2}<a_{1}^{2}+a_{2}^{2}+2 a_{1} a_{2}$. Comparing with the previous, we get $a_{1}^{2}+a_{2}^{2}<2 a_{1} a_{2}$, which is impossible.

An example for $n=4$ can be constructed as follows. Let $a_{1}=a_{2}=1$, choose $a_{3}$ slightly greater than $\sqrt{a_{1}^{2}+a_{2}^{2}}$, and $a_{4}$ slightly greater than $\sqrt{a_{2}^{2}+a_{3}^{2}}$. For example, the values $1 ; 1 ; 1.5 ; 1.9$ work.

## Answer

4.