3 Let $X=\{1,2,3, \cdots, 20\}, A$ be a subset of $X$, and for any $x<y<z, x, y, z \in A$, there exists a triangle with side lengths $x, y, z$. Find the maximum value of $|A|$.

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3. First, let $A_{1}=\{1,2,3,5,8,13\}, A_{2}=\{4,6,10,16\}, A_{3}=\{7$, $12,19\}, A_{4}=\{9,11,20\}, A_{5}=\{14,15\}, A_{6}=\{17,18\}$, then any 3 numbers in $A_{i}$ do not form a triangle, so $A$ can contain at most 2 numbers from $A_{i}$, thus $|A| \leqslant 2 \times$ $6=12$. But if $|A|=12$, then $14,15,17,18 \in A$, thus, $1,2,3,5 \notin A$, so 8 , $13 \in A$, thus, $7,9 \notin A$, so $12,19,11,20 \in A$. But $8+12=20$, contradiction. Therefore, $|A| \neq 12$, so $|A| \leqslant 11$. Next, let $A=\{10,11,12, \cdots, 20\}$, then $A$ meets the requirement. Hence, the maximum value of $\downharpoonleft A \mid$ is 11.