Olympiad Problems: a Variety of Types and Solutions

Abstract

This article provides a methodological analysis and presents solution algorithms for non-standard mathematical problems proposed during the distance (correspondence) rounds of the All-Ukrainian Olympiad at Taras Shevchenko National University of Kyiv. The study explores a diverse range of problem types, including symmetric systems of higher-degree equations, problems involving integer sequences, parametric inequalities, and complex cases in planimetry and solid geometry. The materials are intended for high school students, mathematics educators, and methodologists involved in preparing students for intellectual competitions and admission to higher education institutions with a mathematical profile.