2. Let $\Gamma(S)$ denote the sum of all elements in a non-empty set of integers $S$. Let $A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be a set of positive integers, with $a_{1}<a_{2}<\cdots<a_{n}$. If for every positive integer $n \leqslant 1500$, there exists a subset $S$ of $A$ such that $\Gamma(S)=n$. Find the minimum value of $a_{10}$ that satisfies the above requirement.
(21st United States Olympiad Problem)
2. Let $S_{k}=a_{1}+a_{2}+\cdots+a_{k}(1 \leqslant k \leqslant 11)$, by the problem's condition, there exists an index $m$, such that $S_{m-1}S_{k-1}+1$, and $a_{1}+\cdots+a_{k-1}=S_{k-1}$. This way, there does not exist $S \varsubsetneqq A$, such that $\delta(S)=S_{k-1}+1$. Therefore, $S_{k}=S_{k-1}+a_{k} \leqslant 2 S_{k-1}+1$.

By the problem's condition, $S_{1}=a_{1}=1$, thus (1) using mathematical induction, we can prove that $S_{k} \leqslant 2^{k}-1(1 \leqslant k \leqslant m)$. From $1500 \leqslant S_{m} \leqslant 2^{m}-1$, we get $m=11$. From (1), we have $\frac{S_{k+1}-1}{2} \leqslant S_{k} \leqslant 2^{k}-1(k=1,2, \cdots, 10)$. Since $S_{k}$ is a natural number, therefore, $\left\lceil\frac{S_{k+1}-1}{2}\right\rceil \leqslant S_{k} \leqslant 2^{k}-1$ (where $\lceil x\rceil$ represents the smallest integer greater than or equal to $x$). In particular, $S_{10} \geqslant\left\lceil\frac{S_{11}-1}{2}\right\rceil \geqslant \left\lceil\frac{1500-1}{2}\right\rceil=750$. Also, $S_{8} \leqslant 2^{8}-1=255$, thus $2 a_{10}>a_{9}+a_{10}=S_{10}-S_{8} \geqslant 495$. Therefore, $a_{10} \geqslant 248$.

On the other hand, we can construct a set $A$ that meets the requirements, and $a_{10}=248$. Specifically, take $A=\{1,2,4,8,16,32,64,128,247,248,750\}$. Since $255=2^{7}+2^{6}+\cdots+2+2^{0}$, using binary representation, when $n \leqslant 255$, we can find $S \varsubsetneqq\{1,2,4, \cdots, 128\}$, such that $\delta(S)=n$. Thus, when $n \leqslant 255+247=502$, there exists $S \varsubsetneqq\{1,2,4,128,247\}$, such that $\delta(S)=n$; when $n \leqslant 502+248=750$, there exists $S \varsubsetneqq\{1,2, \cdots, 128,247,248\}$, such that $\delta(S)^{7}=n$; when $n \leqslant 750+750=1500$, there exists $S \varsubsetneqq A$, such that $\delta(S)=n$.