一道全国大学生数学竞赛决赛试题的多种解法

Multiple Solutions to a Final-Round Problem from the National College Students Mathematics Competition

摘要: 本文以第七届全国大学生数学竞赛决赛(非数学类)第二大题为例，从六个不同角度给出证明方法。这些方法包括偏导数链式法则法、隐函数定理法、参数方程法、几何直观法、向量分析法以及微分形式法。通过不同方法之间的比较分析，揭示了它们虽路径迥异却思想同源的深层联系——均植根于多元微分学的链式法则与隐函数存在定理，是同一数学本质在不同视角下的呈现。在此基础上，本文进一步探讨了竞赛数学中一题多解对学生空间想象能力、逻辑推理能力和创新思维培养的教育价值。

Abstract: This paper focuses on the second question (a typical problem concerning tangent planes) from the final round of the 7th National College Students Mathematics Competition (non-mathematics category) and presents proofs from six different perspectives. These approaches include the partial-derivative chain rule method, the implicit function theorem method, the parametric equation method, the geometric intuition method, the vector analysis method, and the differential forms method. Through a comparative analysis of these methods, the paper reveals their deep underlying connection: although they follow different routes, they are rooted in the same mathematical ideas, namely the chain rule in multivariable calculus and the existence theorem for implicit functions. In essence, they represent the same mathematical nature from different viewpoints. On this basis, the paper further discusses the educational value of multiple solutions to a single problem in competition mathematics, particularly in cultivating students’ spatial imagination, logical reasoning ability, and innovative thinking.