In this article, we solve some of the geometry problems of the N\'aboj 2023 competition with the help of a computer, using examples that the software tool GeoGebra Discovery can calculate. In each case, the calculation requires symbolic computations. We analyze the difficulty of feeding the problem into the machine and set further goals to make the problems of this type of contests even more tractable in the future.

We address, through the automated reasoning tools in GeoGebra Discovery, a problem from a regional phase of the Austrian Mathematics Olympiad 2023. Trying to solve this problem gives rise to four different kind of feedback: the almost instantaneous, automated solution of the proposed problem; the measure of its complexity, according to some recent proposals; the automated discovery of a generalization of the given assertion, showing that the same statement is true over more general polygons than those mentioned in the problem; and the difficulties associated to the analysis of the surprising and involved high number of degenerate cases that appear when using the LocusEquation command in this problem. In our communication we will describe and reflect on these diverse issues, enhancing its exemplar role for showing some of the advantages, problems, and current fields of development of GeoGebra Discovery.

Automated reasoning in geometry is available in various software tools for several years, mostly in prover packages. In this paper we pay our attention to a non-trivial presence of a geometry prover in the software tool GeoGebra Discovery [4, 5] that aims at reaching secondary schools with its intuitive user interface. Most importantly, we give a report on a STEM/STEAM project that was discussed in a group of prospective mathematics teachers at the Private University College of Education of the Diocese of Linz in Upper Austria during the winter semester 2022/23, in the frame of a course that focuses on exploiting technology in mathematics education (36 students in 2 working groups). This project consisted of several other experiments that were already communicated by the second author. The discussed activity, a detailed study of the movement of a rocking camel, is however, completely new. Also, some major improvements in the underlying software tool (implemented by the second author with a substantial help of the students' feedback), makes it much easier to model similar project setups and conclude mathematical knowledge in an automated way.

See [1] for a general description and references. The goal of the current contribution is to present some ongoing work regarding two different, but related, important improvements of GeoGebra Discovery. One, to visualize the different steps that GG Discovery performs with a given geometric statement until it declares its truth (or failure). Two, to test, through different elementary examples, the suitability of an original proposal to evaluate the interest, complexity or difficulty of a given statement. Let us advance that our proposal involves the notion of syzygy of a set of polynomials. The relevance of showing details about each of the steps performed by our automated reasoning algorithms implemented in GG Discovery is quite evident. In fact, as a consequence of the result in [2], describing the formalization of the arithmetization of Euclidean plane geometry, proofs of geometric statements obtained using algebraic geometry algorithms are also valid on the synthetic geometry realm.

The use of technical media in Austrian mathematics lessons is largely limited to the GeoGebra medium. GeoGebra [5] proved to be a great tool to visualize and analyze classroom problems, but certain tasks like proving geometric facts rigorously by using a visual explanation is not supported by GeoGebra. As an alternative approach, we focus on introducing JGEX [6] as opposed to GeoGebra, specifically in the area of competition tasks. Geometric proofs are no longer an important part of secondary school curriculum in Austria and many other countries. Formerly, however, Euclidean plane geometry and proving more complicated facts was a part of the expected knowledge of secondary level.

ADG is a forum to exchange ideas and views, to present research results and progress, and to demonstrate software tools at the intersection between geometry and automated deduction. The conference is held every two years. The previous editions of ADG were held in Hagenberg in 2021 (online, postponed from 2020 due to COVID-19), Nanning in 2018, Strasbourg in 2016, Coimbra in 2014, Edinburgh in 2012, Munich in 2010, Shanghai in 2008, Pontevedra in 2006, Gainesville in 2004, Hagenberg in 2002, Zurich in 2000, Beijing in 1998, and Toulouse in 1996. The 14th edition, ADG 2023, was held in Belgrade, Serbia, in September 20-22, 2023. This edition of ADG had an additional special focus topic, Deduction in Education. Invited Speakers: Julien Narboux, University of Strasbourg, France "Formalisation, arithmetization and automatisation of geometry"; Filip Mari\'c, University of Belgrade, Serbia, "Automatization, formalization and visualization of hyperbolic geometry"; Zlatan Magajna, University of Ljubljana, Slovenia, "Workshop OK Geometry"

Automated Deduction in Geometry (ADG) is a forum to exchange ideas and views, to present research results and progress, and to demonstrate software tools at the intersection between geometry and automated deduction. Relevant topics include (but are not limited to): polynomial algebra, invariant and coordinate-free methods; probabilistic, synthetic, and logic approaches, techniques for automated geometric reasoning from discrete mathematics, combinatorics, and numerics; interactive theorem proving in geometry; symbolic and numeric methods for geometric computation, geometric constraint solving, automated generation/reasoning and manipulation with diagrams; design and implementation of geometry software, automated theorem provers, special-purpose tools, experimental studies; applications of ADG in mechanics, geometric modelling, CAGD/CAD, computer vision, robotics and education. Traditionally, the ADG conference is held every two years. The previous editions of ADG were held in Nanning in 2018, Strasbourg in 2016, Coimbra in 2014, Edinburgh in 2012, Munich in 2010, Shanghai in 2008, Pontevedra in 2006, Gainesville in 2004, Hagenberg in 2002, Zurich in 2000, Beijing in 1998, and Toulouse in 1996. The 13th edition of ADG was supposed to be held in 2020 in Hagenberg, Austria, but due to the COVID-19 pandemic, it was postponed for 2021, and held online (still hosted by RISC Institute, Hagenberg, Austria), September 15-17, 2021 (https://www.risc.jku.at/conferences/adg2021).

We describe a prototype of a new experimental GeoGebra command and tool Discover that analyzes geometric figures for salient patterns, properties, and theorems. This tool is a basic implementation of automated discovery in elementary planar geometry. The paper focuses on the mathematical background of the implementation, as well as methods to avoid combinatorial explosion when storing the interesting properties of a geometric figure.

The geometry automated theorem proving area distinguishes itself by a large number of specific methods and implementations, different approaches (synthetic, algebraic, semi-synthetic) and different goals and applications (from research in the area of artificial intelligence to applications in education). Apart from the usual measures of efficiency (e.g. CPU time), the possibility of visual and/or readable proofs is also an expected output against which the geometry automated theorem provers (GATP) should be measured. The implementation of a competition between GATP would allow to create a test bench for GATP developers to improve the existing ones and to propose new ones. It would also allow to establish a ranking for GATP that could be used by "clients" (e.g. developers of educational e-learning systems) to choose the best implementation for a given intended use.