12. 聖公會曾肇添中學 Least Optimal Square Packing in a Square A lot of effort has been devoted into solving the famous Square Packing Problem, which investigates the minimum side length of a square container that can pack n unit squares. This involves the search for the most optimal packing for squares. The aim of this research is to investigate an opposite idea to the original problem. We delve into the least optimal packing of squares, i.e., finding the minimum side length of a square container that can contain all configurations of n unit squares. By considering the idea of a rotating container, we have successfully found the solution to the case of two squares. At the end, by studying the classification of intersections between the configuration and the container, as well as harnessing analytical methods, we have found the exact solution to the general case of n squares. 13. 聖保羅男女中學 Generalising Orthocentres of Triangles to Simplices as the Isogonal Conjugates of the Circumcentres In this paper, we have generalised the orthocentre of a triangle as the isogonal conjugate of the circumcentre of a simplex. Along this generalisation, we have also carried two intriguing properties of the orthocentre of a triangle over to higher dimensions, which says that the isogonal conjugate of the circumcentre of a simplex is either the incentre or an excentre of its pedal simplex, and is also the radical centre of the facetal circumhyperspheres of the simplex. To this end, we have extended the scope of isogonal conjugation with respect to simplices to non-interior points through developing new algebraic and geometric characterisations for it. We have also obtained a higher-dimensional analogue of a curious property of isogonal conjugates with respect to triangles, which says that when both a point and its isogonal conjugate with respect to a simplex are projected onto the facets, the projections formed are co-hyperspherical.
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