Title: Solvability of $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$ Show Chinese title

Title: $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$的可解性

Abstract: Suppose $k,x,$ and $b$ are positive integers, and $a$ is a nonnegative integer such that $k=a+b$. In this paper, we will prove $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$ if and only if $x=a=1$. We do this by looking at different cases depending on the values of $x$ and $k$. We use varying techniques to prove the cases, such as direct proof, verification through Maple software, and a proof technique found in Moser's paper. Previous results from Hanson, St\u{a}nic\u{a}, Shanta, Shorey and Nair are also used. Show Chinese abstract

Abstract: 假设$k,x,$和$b$是正整数，$a$是一个非负整数，使得$k=a+b$。在本文中，我们将证明$\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$当且仅当$x=a=1$。 我们通过根据$x$和$k$的不同取值来分析不同的情况。 我们使用不同的技巧来证明这些情况，例如直接证明、通过 Maple 软件进行验证，以及在 Moser 的论文中找到的证明技巧。 还使用了 Hanson、Stănică、Shanta、Shorey 和 Nair 的先前结果。