It's February and we'd love to help you find the right course plan!

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k a February Highlights and 2025 AoPS Online Class Information
jlacosta   0
Feb 2, 2025
We love to share what you can look forward to this month! The AIME I and AIME II competitions are happening on February 6th and 12th, respectively. Join our Math Jams the day after each competition where we will go over all the problems and the useful strategies to solve them!

2025 AIME I Math Jam: Difficulty Level: 8* (Advanced math)
February 7th (Friday), 4:30pm PT/7:30 pm ET

2025 AIME II Math Jam: Difficulty Level: 8* (Advanced math)
February 13th (Thursday), 4:30pm PT/7:30 pm ET

The F=ma exam will be held on February 12th. Check out our F=ma Problem Series course that begins February 19th if you are interested in participating next year! The course will prepare you to take the F=ma exam, the first test in a series of contests that determines the members of the US team for the International Physics Olympiad. You'll learn the classical mechanics needed for the F=ma exam as well as how to solve problems taken from past exams, strategies to succeed, and you’ll take a practice F=ma test of brand-new problems.

Mark your calendars for all our upcoming events:
[list][*]Feb 7, 4:30 pm PT/7:30pm ET, 2025 AIME I Math Jam
[*]Feb 12, 4pm PT/7pm ET, Mastering Language Arts Through Problem-Solving: The AoPS Method
[*]Feb 13, 4:30 pm PT/7:30pm ET, 2025 AIME II Math Jam
[*]Feb 20, 4pm PT/7pm ET, The Virtual Campus Spring Experience[/list]
AoPS Spring classes are open for enrollment. Get a jump on 2025 and enroll in our math, contest prep, coding, and science classes today! Need help finding the right plan for your goals? Check out our recommendations page!

Don’t forget: Highlight your AoPS Education on LinkedIn!
Many of you are beginning to build your education and achievements history on LinkedIn. Now, you can showcase your courses from Art of Problem Solving (AoPS) directly on your LinkedIn profile! Don't miss this opportunity to stand out and connect with fellow problem-solvers in the professional world and be sure to follow us at: https://www.linkedin.com/school/art-of-problem-solving/mycompany/ Check out our job postings, too, if you are interested in either full-time, part-time, or internship opportunities!

Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1
Monday, Feb 3 - May 19
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Prealgebra 1 Self-Paced

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Prealgebra 2 Self-Paced

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Introduction to Algebra B Self-Paced

Introduction to Geometry
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Intermediate: Grades 8-12

Intermediate Algebra
Wednesday, Feb 12 - Jul 23
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Intermediate Counting & Probability
Monday, Feb 10 - Jun 16
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Thursday, Feb 20 - May 8
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MATHCOUNTS/AMC 8 Basics
Tuesday, Feb 4 - Apr 22
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AMC 10 Problem Series
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Sunday, Feb 9 - Mar 2 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)

AMC 12 Problem Series
Sunday, Feb 23 - May 11

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Sunday, Feb 9 - Mar 2 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)

Special AIME Problem Seminar B
Sat & Sun, Feb 1 - Feb 2 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

F=ma Problem Series
Wednesday, Feb 19 - May 7

Programming

Introduction to Programming with Python
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Tuesday, Feb 25 - May 13

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Thursday, Feb 6 - Apr 24

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Introduction to Physics
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Sunday, Feb 9 - Aug 3
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Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
0 replies
jlacosta
Feb 2, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Legendary problem
BR1F1SZ   5
N 4 minutes ago by GreekIdiot
Source: 2024 July Olympiad P9
An evil magician summons three kings, Arthur, Benjamin and Charles, to play his favorite game.

First, the kings choose two functions $ f, g: \mathbb{N} \to \mathbb{N} $ and reveal them to the magician. Then, the magician secretly selects a positive integer $ n $ and tells Arthur the value of $ f(n) $ and Benjamin the value of $ g(n) $.

Next, Arthur and Benjamin take turns responding aloud, starting with Arthur, with one of the following two statements:
[list]
[*]"I can determine the value of $ n $ with certainty."
[*]"I cannot determine the value of $ n $ with certainty."
[/list]
Charles, who listens to their statements, may interrupt at any moment by declaring, "I can determine the value of $ n $ with certainty." If any of them lies or makes a mistake, the magician will notice and curse them. If Charles correctly deduces the value of $ n $, the three kings win. Otherwise, the magician wins.

One day, the kings discover an ancient scroll describing a specific pair of functions $ f $ and $ g $ that guarantees their victory, regardless of the positive integer $ n $ chosen by the magician. When they challenge the magician using this pair of functions, the magician notices their strategy and changes the rules of the game, making Benjamin the first to respond, followed by Arthur.

Suppose that after this change, there exists some value of $ n $ for which the magician can ensure victory. Prove that exactly two values of $ n $ guarantee the magician’s victory.

(Proposed by BrunZo)
5 replies
BR1F1SZ
Feb 20, 2025
GreekIdiot
4 minutes ago
Set and number theory
XAN4   1
N 6 minutes ago by shanelin-sigma
Source: own
Does there exist a set $\mathcal S$ consisting of $n$ integers$,$ such that $\forall a,b\in \mathcal S,$ $\exists c\in\mathcal S$ such that $c^2\mid a(b+c),$ and that $\exists a\in\mathcal S$ such that if the smallest number $x$ in set $\mathcal S,$ $x\nmid a.$
1 reply
XAN4
31 minutes ago
shanelin-sigma
6 minutes ago
D985 : Bigger inequalities ?
Dattier   4
N 23 minutes ago by Dattier
Source: les dattes à Dattier
Let $a,b>0$ distincts reals numbers.

Is it true that $$\dfrac 1{48} \dfrac{(a+b)^2}{a^3b^3}\geq \dfrac 1{a-b}\ln\left(\dfrac {a(b+1)}{b(a+1)}\right)-4\ln\left(\dfrac{\sqrt{a+1}+\sqrt{b+1}}{\sqrt a+\sqrt b}\right)^2\geq \dfrac 1{48} \dfrac{(a+b+2)^2}{(1+a)^3(1+b)^3}$$?
4 replies
Dattier
Jan 12, 2025
Dattier
23 minutes ago
Random fe
Akacool   7
N 23 minutes ago by the_universe6626
$f(xf(x)+y) = x^2 + f(y)$
from positive real to positive reals
7 replies
1 viewing
Akacool
2 hours ago
the_universe6626
23 minutes ago
Floor function problem
Drago119   0
an hour ago
For $n\geq 3$, consider the set:
$$A_n=\left\{\sum_{k=1}^n \left\lfloor \frac{1}{x_k} \right\rfloor : x_1,x_2,\dots,x_n>0; \sum_{k=1}^n x_k=1\right\}$$Prove that every positive integer $m\geq n^2-n+1$ is part of this set.
0 replies
Drago119
an hour ago
0 replies
Some friendship conditions, proving an equality
Pqrq   2
N an hour ago by bin_sherlo
Source: Turkey 2021 IMO TST Problem 2
In a school with some students, for any three student, there exists at least one student who are friends with all these three students.(Friendships are mutual) For any friends $A$ and $B$, any two of their common friends are also friends with each other. It's not possible to partition these students into two groups, such that every student in each group are friends with all the students in the other gruop. Prove that any two people who aren't friends with each other, has the same number of common friends.(Each person is a friend of him/herself.)
2 replies
Pqrq
May 23, 2021
bin_sherlo
an hour ago
Algebra high school
Nomad_from_QZ   3
N 2 hours ago by Nomad_from_QZ
Find all numerical functions defined on the set of rational numbers such that for all rational numbers , the following equality holds:

f(a + b + c) + f(a) + f(b) + f(c) = f(a + b) + f(b + c) + f(c + a) + f(0).

3 replies
Nomad_from_QZ
Yesterday at 8:34 AM
Nomad_from_QZ
2 hours ago
Staircase-brick with 3 steps of width 2 using 12 unit cubes
orl   11
N 2 hours ago by NTguy
Source: IMO Shortlist 2000, C2
A staircase-brick with 3 steps of width 2 is made of 12 unit cubes. Determine all integers $ n$ for which it is possible to build a cube of side $ n$ using such bricks.
11 replies
orl
Aug 10, 2008
NTguy
2 hours ago
Insane geometry 2
XAN4   1
N 2 hours ago by kiyoras_2001
$\mathcal C$ is the nine-point circle of $\triangle{ABC}$. The three lines tangent to $\mathcal C$ outside $\triangle{ABC}$ parallel to the three sides form $\triangle{A'B'C'}$. Prove that the perspective point of $\triangle{A'B'C'}$ and $\triangle{ABC}$ is the feuerbach point of $\triangle{ABC}$.
1 reply
XAN4
3 hours ago
kiyoras_2001
2 hours ago
Another FE on R+
XAN4   0
2 hours ago
Find all $f$ such that $[x-f(f(x)-y)]f(y)=[y-f(f(y)-x)]f(x)$ and $x\times f(x-f(y))=y\times f(y-f(x))$ for all $x,y\in\mathbb{R}^+$.
By the way, can this be strengthened or only done with one of the equations?
0 replies
XAN4
2 hours ago
0 replies
Most Extensive Collection on Methods and Theorems in Olympiads
onyqz   8
N 2 hours ago by ali123456
Source: Own
What's up guys.
I have decided to create a collection of theorems, methods, techniques and principles that are used in maths olympiads.
When I started practicing for olympiads, most people just told me "learn PHP, double counting, tiling problems, Ceva and Menelaus etc. and you are good to go". Well yeah, they are indeed useful, but also quite standard, so it took a lot of work and research in order to learn the techniques that people might not tell you about (keep in mind that not everyone has access to training camps, so studying on your own is your only option).
The attached file has collected all of the theorems and methods I have come across over the past year. Unnamed lemmas that are quite niche are not included.
All of the methods are also ranked on usefulness, difficulty and from which round starting you might need it (see file).

If you have want any theorems or methods (esp. "advanced" ones) added (or if you find any mistake), then please let me know down below; I will update the file once a while. I am also not very familiar with each and every technique, so you can also help out on the rankings (esp. category difficulty that are ranked "?"). ANY HELP IS APPRECIATED

I am certain you will learn something new.
Peace out :-D
8 replies
onyqz
Oct 26, 2024
ali123456
2 hours ago
Floor inequality
Drago119   2
N 2 hours ago by Drago119
If $x_1,x_2,…,x_n>0$, and $x_1+x_2+…+x_n=1$, find the minimum value of:
$$A=\left\lfloor \frac 1{x_1} \right\rfloor+\left\lfloor \frac1{x_2}\right\rfloor+…+\left\lfloor \frac 1{x_n}\right\rfloor$$
2 replies
Drago119
2 hours ago
Drago119
2 hours ago
a round-robin football tournament
nhathhuyyp5c   3
N 2 hours ago by nhathhuyyp5c
In a round-robin football tournament with $n>4$ teams participating, each team earns $3$ points for a win, $1$ point for a draw, and $0$ points for a loss. At the end of the tournament, it is observed that all teams have the same total points.
$\text{a)}$ Prove that there exist 4 teams with the same number of wins, losses, and draws.
$\text{b)}$ Find the smallest positive integer $n$ such that there do not exist $5$ teams with the same number of wins, losses, and draws.




3 replies
nhathhuyyp5c
Yesterday at 2:16 PM
nhathhuyyp5c
2 hours ago
100 Selected Problems Handout
Asjmaj   28
N 2 hours ago by ali123456
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
28 replies
Asjmaj
Dec 31, 2024
ali123456
2 hours ago
equal angles starting with angles bisectors
parmenides51   0
Apr 27, 2019
Source: 2007 Sharygin Geometry Olympiad Correspondence Round P15
In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.
0 replies
parmenides51
Apr 27, 2019
0 replies
equal angles starting with angles bisectors
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G H BBookmark kLocked kLocked NReply
Source: 2007 Sharygin Geometry Olympiad Correspondence Round P15
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parmenides51
30622 posts
#1 • 5 Y
Y by Adventure10, Mango247, Mango247, Mango247, Rounak_iitr
In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.
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