24.1. 2 ** Let $M=\{1,2, \cdots, 2005\}, A$ be a subset of $M$, if for any $a_i, a_j \in A$, $a_i \neq a_j$, an isosceles triangle can be uniquely determined with $a_i, a_j$ as side lengths, find the maximum value of $|A|$.

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Figure b
When $a<b$, $a, b, b$ must form an isosceles triangle. Therefore, two numbers $a, b$ uniquely determine an isosceles triangle with $a, b$ as two of its sides, which is equivalent to $a, a, b$ not forming an isosceles triangle, i.e., $2a < b$. Let $A=\left\{a_{1}<a_{2}<\cdots<a_{n}\right\}$ be a subset of $M$ that meets the conditions, then $2a_{i} \leqslant a_{i+1}$ (spacing estimation). Thus, $2005 \geqslant a_{n} \geqslant 2a_{n-1} \geqslant \cdots \geqslant 2^{n-1}a_{1} \geqslant 2^{n-1}$, so $n \leqslant 11$.

Furthermore, let $A=\{1,2,4, \cdots, 1024\}$, then $|A|=11$. For any $a_{i}, a_{j} \in A$, if $i < j$, then $a_{i}=2^{i-1}, a_{j}=2^{j-1}$, and $2a_{i}=2^{i} \leqslant 2^{j-1}=a_{j}$, thus the only isosceles triangle that can be formed with $a_{i}, a_{j}$ is $\left(a_{i}, a_{j}, a_{j}\right)$, so $A$ meets the conditions.
In summary, the maximum value of $|A|$ is 11.