These two volumes are an ideal resource for advanced physics students preparing to progress to the PhD level, offering a carefully curated selection of intellectually engaging and pedagogically valuable problems spanning the breadth of the subject. Each problem is accompanied by a fully worked, detailed solution designed to illuminate problem-solving strategies, clarify conceptual pitfalls, and highlight key take-home messages. The selection of problems reflects emerging trends in physics pedagogy and topical domains of study. These volumes are as much a training ground for problem-solving habits essential to research careers as they are a resource for examination preparation, emphasizing clarity, strategy, and analytical depth. All problems are taken from the qualifying examinations of the University of California, Santa Cruz Physics Department. While serving as an unmatched resource for students preparing for these specific exams, both volumes remain highly relevant to the wider physics community. Graduate students worldwide preparing for qualifying or comprehensive examinations will find excellent preparation material, while advanced undergraduates can benchmark their readiness for graduate-level study. The problems are ranked by difficulty using a three-tier star system, making the books readily adaptable to students of different levels. Instructors will also find great value in this reliable repository of questions that have been vetted repeatedly under high-stakes conditions. This first volume covers the topics of classical mechanics and mathematical methods for physics. The problems are of a level accessible to advanced undergraduates as well as master's students transitioning to the PhD level.