Models of Sample Spaces in Olympiad Problems

Dragomir Grozev, Stanislav Harizanov
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences (Bulgaria)

https://doi.org/10.53656/math2025-3-1-mss

Abstract. Applications of the probabilistic method in combinatorial problems are shown. In most sources, the construction of the corresponding sample space is natural and trivial and as a result, the existence of certain objects is proven. The emphasis here is on constructing an appropriate sample space. In one of the problems, the process is even reversed – probabilistic properties of objects lead us to construct an example of a probability space – a construction that is actually sought. Three applications of the method are considered – two problems from this year’s Bulgarian competitions: problem 10.3 from the Bulgarian National Olympiad 2025 (Regional Round), and problem 10.3 from the Bulgarian winter math competition 2025 (a generalization of the latter is discussed). We end with problem 5 from 2023 International Math Olympiad. The authors are not aware of a solution to the latter using such an approach.

Keywords: combinatorics; probabilistic method; sample space; probability space; olympiad math problems; Bulgarian National Math Olympiad; International Math Olympiad

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