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PAVOLJOZEF©AFÁRIKUNIVERSITYINKO©ICE FACULTYOFSCIENCE Automated Generation of Planar Geometry Olympiad Problems MASTER’S THESIS Field of Study: Informatics Institute: Institute of Computer Science Supervisor: doc. RNDr. Stanislav Krajči, PhD. Consultant: Mgr. Michal Rolínek, PhD. Ko¹ice 2020 Bc. Patrik Bak Acknowledgments I would like to express my gratitude to: my supervisor Stanislav Krajči, for his support in granting me full academic freedom to pursue my own ideas; my consultant Michal Rolínek, whose expertise and advice took this text to an entirely new dimension; Pavel ©alom, for his immensely careful reading of the entire formal model and problem generation sections; and Dávid Uhrik, for countless remarks on every part of the text in terms of both content and style. Univerzita P. J. Šafárika v Košiciach Prírodovedecká fakulta ZADANIE ZÁVEREČNEJ PRÁCE Meno a priezvisko študenta: Bc. Patrik Bak Študijný program: Informatika (Jednoodborové štúdium, magisterský II. st., denná forma) Študijný odbor: Informatika Typ záverečnej práce: Diplomová práca Jazyk záverečnej práce: anglický Sekundárny jazyk: slovenský Názov: Automated generation of planar geometry olympiad problems Názov SK: Automatické generovanie planimetrických úloh Matematickej olympiády Cieľ: Design and implement software that is able to generate planar geometry olympiad problems by extending an initial configuration with new geometrical objects and subsequently finding non-trivial theorems in the generated configurations. Literatúra: [1] Rajiv Bagai, Vasant Shanbhogue, Jan M. Żytkow, and Shang-Ching Chou. Automatic theorem generation in plane geometry. In: International Symposium on Methodologies for Intelligent Systems. 1993. 415–424. [2] Shang-Ching Chou, Xiao-Shan Gao, and Jing-Zhong Zhang. A deductive database approach to automated geometry theorem proving and discovering. Journal of Automated Reasoning. 2000, 25 (3), 219–246 [3] Andreas Poulos. A research on the creation of problems for mathematical competitions. Teaching of Mathematics. 2017, 20 (1) Vedúci: doc. RNDr. Stanislav Krajči, PhD. Konzultant: Mgr. Michal Rolínek, PhD. Oponent: RNDr. Ondrej Krídlo, PhD. Ústav : ÚINF - Ústav informatiky Riaditeľ ústavu: RNDr. Ondrej Krídlo, PhD. Dátum schválenia: 02.04.2020 Abstract We present a system that generates planar geometry problems suitable for mathematical competitions such as the International Mathematical Olympiad. Our solution consists of the following assets: (1) a novel, provably correct, problemgeneration algorithm, (2) bulk filtering algorithms of easy problems in- spired by geometry theorem proving methods, primarily the deductive database # method, (3) a novel problem quality ranking algorithm. A C implementation of the system has been developed and made available on GitHub https:// github.com/PatrikBak/GeoGen.Thedevelopedalgorithmshavebeentestedin manysmallandlarge-scale experiments that generated, rated, and sorted thou- sands of problems, nine of which are enclosed in the appendix. Five generated problems have been proposed to the International Mathematical Olympiad, whilst two generated problems have already been accepted to the Czech-Slovak mathematical contests. Keywords: geometry problem generation, geometry theorem proving, mathe- matical olympiad problems, planar geometry problems
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