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# 恒隆數學獎

1. 拔萃男書院 An Investigation on the Rings of Integer-Valued Polynomials on Gaussian Integers and Integer-Valued Continuous Functions This is an investigation on the ring of integer-valued polynomials on the Gaussian integers and the r ing of integer-valued continuous function on rational integers, inspired by the results from integer-valued polynomials on the rational integers. Polynomials in the first ring map Gaussian integers to Gaussian integer values while functions in the second ring map rational integers to rational integers. This investigation explores their properties as rings, following a chain of class inclusions, which includes the most commonly known domains. The properties of rings of polynomials over algebraic integers, continuously differentiable functions on rational integers, and continuous functions on Gaussian integers are also discussed. 2. 拔萃男書院 On the Basel Problem: Generalizations to Other Power Series The Basel problem is about finding the sum of the reciprocals of all perfect squares. This problem is first posed by Pietro Mengoli in 1650 and was solved by Leonhard Euler in 1734. Euler proved that the sum of the series is π2/6. In this report, inspired by an idea suggested by the YouTube channel 3blue1brown in 2018, we attempt to give a new proof to the Basel problem. After that, we discuss some possible generalizations of the Basel problem, by finding the sum of reciprocals of squares and cubes of the form an+b. Furthermore, we discuss how the sum of reciprocals of integral powers of an+b can be computed, and the relation between ζ(3) and the results we have achieved. 3. 拔萃男書院 On the Variations of the Brachistochrone Curve In this research report we will provide our own derivation of the classical Brachistochrone Curve. Afterwards, we will study some variations of the Brachistochrone Problem with two gravitational forces, friction, and finally some concepts from non-Newtonian mechanics. 2021 年恒隆數學獎入圍隊伍專題研究報告摘要 (只有英文版本) (排名以學校英文名稱順序)

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