13. St. Paul’s Co-educational College 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. 14. St. Paul’s College Covering 45 Points Configuration with Disjoint Unit Disks We study the conjecture of the non-existence of disjoint unit disks covering the 45 points configuration, proposed by Aloupis, Hearn, Iwasawa, and Uehara. We proved that if such covering exists, then the number of disks N used must be 3 ≤ N ≤ 5. We also showed that if N = 3, then the number of points covered in the outermost circle must be (8, 7, 6) up to permutation. 15. Wah Yan College, Kowloon On the Littlewood Problem and Sum of Two Squares in the Ring Zq Sum of two squares is a well-developed topic in number theory and J.E. Littlewood asked whether, for given unequal positive numbers h and k, there exist infinitely many n such that n, n + h and n + k are sums of two squares. Mong, Lai, and Mak showed a method to find such triples but n must be a perfect square and one of h, k, and | h − k| must also be a perfect square. In this paper, we developed an explicit formula for generating infinitely many n which need not be a perfect square such that n, n + 1 , and n + 3 are sums of two squares. We also developed a formula to solve Littlewood Problem where there is no restriction on h or k and this result is found to be related to sum of two squares among Eisenstein integers. Lastly, we looked at the cyclic ring Zq: Harrington, Jones and Lamarche found the condition when not all elements in Zq are sums of two nonzero squares. We found the number of elements which are sums of two nonzero squares in all such cases.

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