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
Sunday, Mar 2 - Jun 22
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Sunday, Apr 13 - Aug 10

Prealgebra 1 Self-Paced

Prealgebra 2
Sunday, Feb 16 - Jun 8
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Prealgebra 2 Self-Paced

Introduction to Algebra A
Sunday, Feb 16 - Jun 8 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)
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Introduction to Algebra A Self-Paced

Introduction to Counting & Probability
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Wednesday, Apr 16 - Jul 2

Introduction to Counting & Probability Self-Paced

Introduction to Number Theory
Sunday, Feb 16 - May 4
Monday, Mar 17 - Jun 9
Thursday, Apr 17 - Jul 3

Introduction to Algebra B
Thursday, Feb 13 - May 29
Sunday, Mar 2 - Jun 22
Monday, Mar 17 - Jul 7
Wednesday, Apr 16 - Jul 30

Introduction to Algebra B Self-Paced

Introduction to Geometry
Friday, Feb 14 - Aug 1
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Wednesday, Apr 23 - Oct 1

Intermediate: Grades 8-12

Intermediate Algebra
Wednesday, Feb 12 - Jul 23
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
Monday, Apr 21 - Oct 13

Intermediate Counting & Probability
Monday, Feb 10 - Jun 16
Sunday, Mar 23 - Aug 3

Intermediate Number Theory
Thursday, Feb 20 - May 8
Friday, Apr 11 - Jun 27

Precalculus
Tuesday, Feb 25 - Jul 22
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Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21

Calculus
Friday, Feb 28 - Aug 22
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Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tuesday, Feb 4 - Apr 22
Sunday, Mar 23 - Jun 15
Wednesday, Apr 16 - Jul 2

MATHCOUNTS/AMC 8 Advanced
Sunday, Feb 16 - May 4
Friday, Apr 11 - Jun 27

AMC 10 Problem Series
Sunday, Feb 9 - Apr 27
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23

AMC 10 Final Fives
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

AMC 12 Final Fives
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
Sunday, Feb 16 - May 4
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Intermediate Programming with Python
Tuesday, Feb 25 - May 13

USACO Bronze Problem Series
Thursday, Feb 6 - Apr 24

Physics

Introduction to Physics
Friday, Feb 7 - Apr 25
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Physics 1: Mechanics
Sunday, Feb 9 - Aug 3
Tuesday, Mar 25 - Sep 2

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
USAPHO vs USAMO
cookedflatpann   3
N a minute ago by remedy
How does usapho honorable mention compare with USAMO qual in prestige? I just messed up the AIME after 140+ AMC and USAMO wont be happening on time for college apps. shifting focus to usapho, so the question came up. thoughts?
3 replies
cookedflatpann
Feb 6, 2025
remedy
a minute ago
exponential diophantine with factorials
skellyrah   2
N 5 minutes ago by polishedhardwoodtable
find all non negative integers (x,y) such that $$ x! + y! = 2025^x + xy$$
2 replies
skellyrah
Yesterday at 6:25 PM
polishedhardwoodtable
5 minutes ago
100 Selected Problems Handout
Asjmaj   31
N 32 minutes ago by BGR2025
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
31 replies
Asjmaj
Dec 31, 2024
BGR2025
32 minutes ago
Recursive Grid Construction
john0512   13
N 33 minutes ago by Ritwin
Source: 2023 USA TSTST Problem 3
Find all positive integers $n$ for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly $n$ cells (and whose edges lie along the lines of the grid) contains an odd number of red cells.

Proposed by Merlijn Staps
13 replies
john0512
Jun 26, 2023
Ritwin
33 minutes ago
Geometry with fix circle
falantrng   30
N 36 minutes ago by Ilikeminecraft
Source: RMM 2018 Problem 6
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.
30 replies
falantrng
Feb 25, 2018
Ilikeminecraft
36 minutes ago
Equation has no integer solution.
Learner94   32
N 39 minutes ago by polishedhardwoodtable
Source: INMO 2013
Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.
32 replies
Learner94
Feb 3, 2013
polishedhardwoodtable
39 minutes ago
Stronger form of ISL 2013. C3.
Natrium   1
N an hour ago by mathelvin
Source: IMO Shortlist 2013. C3.
A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time.
(i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it.
(ii) At any moment, he may double the whole family of imons in the lab by creating a copy $I'$ of each imon $I$. During this procedure, the two copies $I'$ and $J'$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I'$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment.

Prove that the physicist may apply a sequence of such operations, using (ii) at most once, resulting in a family of imons, no two of which are entangled.
1 reply
Natrium
Nov 10, 2024
mathelvin
an hour ago
Conan tries to beat the detective from the west
iStud   1
N an hour ago by iStud
Source: KTOM
Conan and Heiji play a board game. In turn, they write natural numbers that are not greater than $n$, starting from Conan. The numbers that are already written on the board before can't be written again. A player wins if after his turn is done, there are $3$ numbers on the board that form arithmetic sequence or geometry sequence. Prove that there are at least $1533$ different values of $n$ from $3\le n\le 2023$ so that Conan has a winning strategy.
1 reply
iStud
Yesterday at 3:52 PM
iStud
an hour ago
IMO ShortList 2002, algebra problem 1
orl   125
N 2 hours ago by ali123456
Source: IMO ShortList 2002, algebra problem 1
Find all functions $f$ from the reals to the reals such that

\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]

for all real $x,y$.
125 replies
orl
Sep 28, 2004
ali123456
2 hours ago
Prove that OA and RA are perpendicular
MellowMelon   87
N 2 hours ago by Maximilian113
Source: USA TSTST 2011/2012 P4
Acute triangle $ABC$ is inscribed in circle $\omega$. Let $H$ and $O$ denote its orthocenter and circumcenter, respectively. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Rays $MH$ and $NH$ meet $\omega$ at $P$ and $Q$, respectively. Lines $MN$ and $PQ$ meet at $R$. Prove that $OA\perp RA$.
87 replies
MellowMelon
Jul 26, 2011
Maximilian113
2 hours ago
easy modified cauchy fe
blueprimes   3
N 3 hours ago by jasperE3
Source: OTIS Z2A3B740
Determine all functions $f:\mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ such that
\[ f(x) + f(y) + 2xy = f(x + y) \]for all nonnegative real numbers $x$ and $y$.
3 replies
blueprimes
5 hours ago
jasperE3
3 hours ago
Prove two circles intersect on line BC
62861   71
N 3 hours ago by ihatemath123
Source: USA Winter TST for IMO 2019, Problem 1 and TST for EGMO 2019, Problem 2, by Merlijn Staps
Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$.

Merlijn Staps
71 replies
62861
Dec 10, 2018
ihatemath123
3 hours ago
Functional Equation
IstekOlympiadTeam   12
N 3 hours ago by ali123456
Source: Baltic Way 2015
Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the equation \[|x|f(y)+yf(x)=f(xy)+f(x^2)+f(f(y))\]for all real numbers $x$ and $y$.
12 replies
IstekOlympiadTeam
Nov 8, 2015
ali123456
3 hours ago
Collinearity with orthocenter
liberator   176
N 3 hours ago by Maximilian113
Source: IMO 2013 Problem 4
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand
176 replies
liberator
Jan 4, 2016
Maximilian113
3 hours ago
concurrent starting with an inscribed ABCD and tangents from a point on AC
parmenides51   1
N Feb 15, 2020 by Mathematicsislovely
Source: 2014 IMAC Arhimede P2
A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.
1 reply
parmenides51
May 2, 2019
Mathematicsislovely
Feb 15, 2020
concurrent starting with an inscribed ABCD and tangents from a point on AC
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Source: 2014 IMAC Arhimede P2
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parmenides51
30625 posts
#1 • 2 Y
Y by Mathematicsislovely, Adventure10
A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.
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Mathematicsislovely
245 posts
#2 • 1 Y
Y by Adventure10
Obviously,$BDCA$ and $CDEA$ are harmonic quadrilateral.So $(B,D;C,A)=-1$ $(C,D,E,A)=-1$. Now,Let $BE \cap AD=M$.We will prove that CQ will pass through $M$.
$BC \cap AD=N$
Now $-1=(C,D;E,A)=B(C,D;E,A)=(N,D;M,A)$
And $-1=(B,D;C,A)=C(B,D;C,A)=(N,D;CC \cap AD, A)$.So $CQ \cap AD= M$. So $AD$,$BE$,$CQ$ are concurrent.
This post has been edited 2 times. Last edited by Mathematicsislovely, Feb 15, 2020, 5:50 PM
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