This study aims at analysing students’ mistakes in solving mathematical olympiad problems. The test was conducted in SMA Negeri 1 Palembang involving 40 students. The main topics are algebra, combinatorics, geometry, and number theory. The analysis is based on Newman Error Analysis. The results show that the most common mistakes are found in combinatorics and geometry. The most common mistakes analysed using Newman Error are comprehension and transformation errors. In order to solve this problem, teachers need to assign various mathematical problems. Hence, the students will have more flexibility to apply problem-solving strategies.

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Proof-based learning is learning mathematics through proof and proving to strengthen students' concepts. The use of APOS theory (Action, Process, Object, and Schema) aims to describe students' mental structures summarized in Hypothetical Learning Trajectory (HLT). Their interest in learning mathematics requires students to have reasoning ability. This study aims to produce a mini theory of proof-based learning using the APOS theory approach about mental structure and mental mechanisms. The role of learning is to improve students' reasoning ability. The research method used is the design research validation type. The research subjects were 34 students from SMA Negeri 1 Palembang. For this reason, this study will discuss the construction of HLT which will become a mini theory. In this paper, the researcher will focus on the first lesson’s HLT proof-based learning using APOS theory approach. This study's results are to compare HLT and ALT, develop proof learning, and test the truth of the hypotheses made based on the research methodology.