Here is a problem I saw a month ago. This is an inequality that can be interpreted as an inequality between the expectation of a random variable and the expectation of the sum of two independent copies of a this random variable. Problem (Romanian TST 2006, d2,p4). Let be real numbers. Prove that (Dan Schwartz, a discrete version of Putnam 2003, p6) This inequality can be written in this more convinient form and it appeared like that in an Iranian TST 2006 – see [3]. We divide both sides by and get Let and be two independent identically distributed (iid) random variables each of which ta ..read more

Here is a nice problem from Balkan Math Olympiad 2023 Shortlist. The main point is to establish a negative result about some process. We have a team (of girls or princesses) that works together in order to achieve its goal of saving as many boys (princes) as possible. We have to prove that they cannot do it better than some threshold. It’s instructive to see the method of establishing negative results in this spirit. I would call it the method of cheating. Actually, it’s the same idea as in the famous problem about the hunter and the rabbit – IMO 2017, problem 3 – see [2] and [3]. Since the st ..read more

In the previous post I considered a combinatorial problem from Balkan Math Olympiad (BMO) 2023 shortlist. Here we continue with a problem from the same shortlist – a problem that I proposed to BMO 2023. Problem (BMO SL 2023, A4). Prove that there exists a real number such that for each sequence , there are infinitely many pairs such that 2) Prove that the above statement holds for (Dragomir Grozev) This was the exact wording I proposed to BMO 2023. The text that appeared in the shortlist was changed a bit. It wants to show that it holds for some constant . In fact, the official solution in ..read more

Here is a nice combinatorial problem that was given in the Bulgarian team selection tests for RMM 2024. It was from BMO 2023 shortlist Problem (BMO 2023 Shortlist, C3). In a given community of people, each person has at least two friends within the community. Whenever some people from the community sit on a round table such that each adjacent pair of people are friends, it happens that no non-adjacent pair of people are friends. Prove that there exist two people of this community such that each has exactly two friends and they have at least one common friend. I’ll present two solutions – the o ..read more

Here is an interesting geometrical problem given in the recently held Bulgarian National Math Olympiad (NMO) 2024. It is actully a converse version of a well-known fact, which makes it curious. Here it is. Problem (Bulgarian NOM 2024, p2). A triangle is given. The points and are respectively on and such that and Let and Find (Nikolay Nikolov, Aleksander Ivanov) Solution. Answer: Denote . We interpret the configuration as follows see fig. 1. We have an isosceles triangle with and Initially Then, the three points start moving with equal velocities, that is, It ends when reaches ..read more

I want to comment here on a problem I gave to Bulgarian TST for BMO 2024. Here is the story of this problem. About two years ago I came across a problem, (I don’t remember it’s statement – it was about a bipartite graph) and my idea was to delete its edges one by one, so that when one deletes an edge incident to a vertex, one has to delete another edge incident to the same vertex within a certain number of steps, say . It was interesting to estimate the minimum (as a function of the number of vertices ). Of course it depends on the given graph. For example, if you have a complete graph of ve ..read more

Two nice number theory (NT) flavored problems were given at the recent RMM (Romanian Master of Mathematics) competition. The only needed knowledge of NT that one has to have in order to solve them consists of the following two facts. For any prime 1) If then 2) For any integers with we have or Yes, a fifth grader knows it. Still the problems are not so easy. Some people want strict boundaries between different subjects in math competitions. They want for example some “pure” NT problems. Of course, mathematics has different subjects. But they share common methods and technique called Mat ..read more

Here we consider problem 6 of the recent Romanian Master of Mathematics competition – a very difficult problem. Nobody solved it during the competition. Let me point out a very similar problem – RMM Shortlist 2020, A1 – see [3], proposed by the same author. I think it’s of similar difficulty. In my opinion both are not suitable for a high school Olympiad level, although I like them because it’s real math. Maybe they fit for Miklos Schweitzer competition, but on the other hand both problems are known results – the current one by Filaseta and Konyagin – see [2] – so the papers can easily be foun ..read more

This is a continuation of the previous post. I’ll show a more elegant way that will allow us to find all solutions to the functional equation we considered, but it requires a tool called the Krein – Milman theorem. The outline of this post is to first recall the problem, then Krein – Milman theorem and the intuitive motivation behind it. We will finally apply it. Problem (China MO). Let be a function satisfying Prove that In the previous post – see [1] – we proved it using finite differences and making some estimates. Krein – Milman Theorem. Let be a convex set inside some real vector spac ..read more

A year or two ago I saw an interesting problem. The main point was to prove an estimate for a function that has a kind of mean value property – it’s defined on the 2-dimensional lattice grid and its value at each lattice point equals the average of its values on some neighboring knots. It had no solution, just a vague hint. I think it’s still unsolved in AoPS forum, but unfortunately, I lost the link and I lack patience to search that site. It was given at some Chinese Olympiad or some selection test, not sure. It is a tough problem, although the statement is just one short sentence, and it se ..read more