Let ,such that Prove that :

In triangle , prove that

Find all positive integers with the following property: the positive divisors of have a permutation such that for , the number is a perfect square.

In a company of people some pairs are enemies. A group of people is called unsociable if the number of members in the group is odd and at least , and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most unsociable groups, prove that it is possible to partition the company into parts so that no two enemies are in the same part.

Proposed by Russia

There are two bins A and B which contain 12 balls and 24 balls, respectively. Each of these balls is marked with one letter: X, Y, or Z. In each bin, each ball is equally likely to be chosen. Randomly picking from bin A, the probability of choosing balls marked X and Y are and , respectively. Randomly picking from bin B, the probability of choosing balls marked X and Y are and , respectively. If the contents of the two bins are merged into one bin, what is the probability of choosing two balls marked X and Y from this bin?

Happy New Year to all AoPSers!
 :clap2:

Here’s my modest gift to you all. Although I haven’t been very active in the forums, the AoPS community contributed to an immense part of my preparation and left a huge impact on me as a person. Consider this my way of giving back. I also want to take this opportunity to thank Evan Chen—his work has consistently inspired me throughout my olympiad journey, and this handout is no exception.



With 2025 drawing near, my High School Olympiad career will soon be over, so I want to share a compilation of the problems that I liked the most over the years and their respective detailed write-ups. Originally, I intended it just as a personal record, but I decided to give it some “textbook value” by not repeating the topics so that the selection would span many different approaches, adding hints, and including my motivations and thought process.

While IMHO it turned out to be quite instructive, I cannot call it a textbook by any means. I recommend solving it if you are confident enough and want to test your skills on miscellaneous, unordered, challenging, high-quality problems. Hints will allow you to not be stuck for too long, and the fully motivated solutions (often with multiple approaches) should help broaden your perspective. 



This is my first experience of writing anything in this format, and I’m not a writer by any means, so please forgive any mistakes or nonsense that may be written here. If you spot any typos, inconsistencies, or flawed arguments whatsoever (no one is immune :blush: ), feel free to DM me. In fact, I welcome any feedback or suggestions.

I left some authors/sources blank simply because I don’t know them, so if you happen to recognize where and by whom a problem originated, please let me know. And quoting the legend: “The ideas of the solution are a mix of my own work, the solutions provided by the competition organizers, and solutions found by the community. However, all the writing is maintained by me.” 



I’ll likely keep a separate file to track all the typos, and when there’s enough, I will update the main file. Some problems need polishing (at least aesthetically), and I also have more remarks to add.

This content is only for educational purposes and is not meant for commercial usage.



This is it! Good luck in 45^2, and I hope you enjoy working through these problems as much as I did!

Here's a link to Google Drive because of AoPS file size constraints: Selected Problems

Let be reals such that Prove thatLet be reals such that Prove that

Does there exist a convex polyhedron with an odd number of sides, where each side is centrally symmetric?

Let and Prove that

Let n>1 be odd. A row of n spaces is initially empty. Alice and Bob alternate moves (Alice first); on each turn a player may either
1. Place a stone in any empty space, or
2. Remove a stone from a non-empty space S, then (if they exist) place stones in the nearest empty spaces immediately to the left and to the right of S.

Furthermore, no move may produce a position that has appeared earlier. The player loses when they cannot make a legal move.
Assuming optimal play, which move(s) can Alice make on her first turn?