9. 葵涌蘇浙公學 On Solutions of the Exponential Diophantine Equation px − qy = z2 In this paper, solutions of the exponential Diophantine equation p x − q y = z 2 are investigated. In fact, using modular arithmetic with new tricks, we are able to prove Theorem 1.1 that it has at most one solution x, y, z ∈ N, where p ≡ 3 (mod 4) and q ≡ 1 (mod 4). Meanwhile, our Theorem 1.2 concerns the special case when p ≡ 1 (mod 4) and q = 5. We prove in part (a) that it has at most one solution provided that p satisfies p ≡ 3, 7 (mod 10). Besides, part (b) guarantees that the solution of the particular equation 13x − 5y = z2 is unique and this fills the loophole of Burshtein’s proof. Alternatively, another proof of this result is given and the main tools we employ are results from linear forms in logarithms of algebraic numbers developed by Baker and Davenport. Interestingly, with applications of primitive Pythagorean triples, we reveal in part (c) a new connection between solutions with even y and classical half-generalized Fermat numbers p. 10. 保良局百周年李兆忠紀念中學 Generalization of Stern’s Diatomic Sequence Stern’s diatomic sequence is defined as a1 = 1, a2k = ak and a2k+1 = ak + ak+1. It has many useful properties such as { an / an+1 }n⩾1 that runs through all positive rational numbers exactly one time. In this research, we generalize the coefficient of ak and ak+1, which can be any real number p, r, and s, and find 2 closed forms and some summation formulas of it. We also investigate the properties in number theory when p, r, and s are positive integers. 11. 香港培正中學 Mean Shadow of Rotating Objects In this paper, we conduct an analysis of the problem concerning the mean shadow cast by rotating objects. The original problem was introduced by Cauchy in 1832. He proposed solutions for the 2-D and 3-D scenarios in 1842 and 1850, respectively. In the original problem, the shadow was formed by orthogonal projection. In 2022, the problem was revisited under the 3-D scenario of a light source with finite distance above the rotation center. Instead of 3-D scenario, we focus on the 2-D case and generalize the problem by placing the light source arbitrarily. We derive explicit formulae of the mean shadow. With these formulae, we provide a numerical method to compute the mean shadow, which surpasses the conventional simulation.
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