10. 聖公會曾肇添中學 On the Generalization of the Triangle Peg Problem to Higher Dimensions A lot of work has been put into solving special cases of the famous Square Peg Problem, which focuses on the two-dimensional space. The aim of this project is to investigate the generalization of the Triangle Peg Problem into manifolds of higher dimensions. By considering the Triangle Peg Problem, we have successfully proven the Tetrahedron Peg Problem, i.e. for every smooth compact connected surface, there exists four distinct points on the surface which can form a regular tetrahedron. Finally, by induction, we have generalized a variant of the Triangle Peg Problem to even higher dimensions. 11. 聖保羅男女中學 Solvability of the General Pell’s Equation, Quadratic Residuosity, and Real Quadratic Fields of Class Number Two A new and practical test for determining the solvability of the general Pell's equation x2 − Dy2 = n will be offered through proving one necessary condition and one sufficient condition for this renowned quadratic Diophantine equation to be solvable in integers. The test involves only prime factorization and checking of certain simple quadratic residuosity relations. While the necessary condition will be comparatively more straightforward, the sufficient condition in a form of conditional converse of the former will require algebraic number theory tools to formulate and analyze. To prove this sufficient condition, the solvability in question will be transformed into the question of principality of certain well-designed ideal class in a real quadratic field Q( D) of class number two.