and 
be sequences of real numbers for which 
and
for all positive integers 
.
for which 
for all 
)
and 
be two concentric circles. Let 
and 
be any two equilateral triangles inscribed in 
and 
are any two points on ![\[ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. \]](http://latex.artofproblemsolving.com/2/a/7/2a73de753f0332329b743d494272d3fb85f60ef9.png)
(meaning 
takes pairs of positive real numbers to positive real numbers) such that for any real numbers 
,
holds; and
then 
.
Prove that
Y by BaishuakRayimbek, Trenod, Ender_Claddy217, menpo, erke667, farbyy, Abidabi, megarnie, cursed_tangent1434, ehuseyinyigit, NicoN9, Ab_Rin, Dattier, Funcshun840, Pekiban, onyqz, Oksutok, GeorgeRP, Om245, happypi31415, Tem8, Jack_w, jason02, GeoKing, YaoAOPS, Aryan27, R8kt, anantmudgal09, Supercali, TestX01, Sedro, math_comb01, g0USinsane777, Vivouaf, TensorGuy666, CyclicISLscelesTrapezoid, devrandom42, ohhh, internationalnick123456, cubres, Iveela, yobu, hectorleo123, zhenghua, EpicBird08, Rounak_iitr
Happy New Year to all AoPSers!
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
), 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
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
), 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
