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
Friday, Mar 28 - Jul 18
Sunday, Apr 13 - Aug 10

Prealgebra 1 Self-Paced

Prealgebra 2
Sunday, Feb 16 - Jun 8
Tuesday, Mar 25 - Jul 8
Sunday, Apr 13 - Aug 10

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)
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28

Introduction to Algebra A Self-Paced

Introduction to Counting & Probability
Sunday, Feb 9 - Apr 27 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)
Sunday, Mar 16 - Jun 8
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
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
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
Sunday, Mar 16 - Aug 24
Wednesday, Apr 9 - Sep 3

Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21

Calculus
Friday, Feb 28 - Aug 22
Sunday, Mar 30 - Oct 5

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
Monday, Mar 24 - Jun 16

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
Sunday, Mar 30 - Jun 22

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
Subsets disjoint with colours
Twoisaprime   12
N a minute ago by DottedCaculator
Source: RMM 2025 P6
Let $k$ and $m$ be integers greater than $1$. Consider $k$ pairwise disjoint sets $S_1,S_2, \cdots S_k$; each of these sets has exactly $m+1$ elements, one of which is red and the other $m$ are all blue. Let $\mathcal{F}$ be the family of all subsets $F$ of $S_1 \bigcup S_2\bigcup \cdots S_k$ such that, for every $i$ , the intersection $F \bigcap S_i$ is monochromatic; the empty set is also monochromatic. Determine the largest cardinality of a subfamily $\mathcal{G} \subseteq \mathcal{F}$, no two sets of which are disjoint.
Proposed by Russia, Andrew Kupavskii and Maksim Turevskii
12 replies
Twoisaprime
Feb 13, 2025
DottedCaculator
a minute ago
Incenters, Excenter and Circumcenter
Retemoeg   0
2 minutes ago
Source: Polish MO - 2nd round 2017
Let $\triangle ABC$ be a scalene triangle with incenter $I$ and $J$ the $A$-excenter. $CA, AB$ touches $(I)$ at $E, F$, respectively. Denote $M, N$ the midpoints of segments $JF, JE$. Show that $BM$ meets $CN$ on the circumcircle of $\triangle ABC$.
0 replies
Retemoeg
2 minutes ago
0 replies
IMOC 2019 G5 (DH bisects <EDF, circumcircle, orthocenter related)
parmenides51   5
N 21 minutes ago by Retemoeg
Source: https://artofproblemsolving.com/community/c6h1958463p13536764
Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$. The exterior angle bisector of $\angle BAC$ intersects circumcircle of $\vartriangle ABC$ at $N \ne  A$. Let $D$ be another intersection of $HN$ and the circumcircle of $\vartriangle ABC$. The line passing through $O$, which is parallel to $AN$, intersects $AB,AC$ at $E, F$, respectively. Prove that $DH$ bisects the angle $\angle EDF$.
IMAGE
5 replies
parmenides51
Mar 22, 2020
Retemoeg
21 minutes ago
Romanian NMO 2021 inequality
soryn   10
N 23 minutes ago by SomeonesPenguin
If $a,b,c>0,a+b+c=1$,then:

$\frac{1}{abc}+\frac{4}{a^{2}+b^{2}+c^{2}}\geq\frac{13}{ab+bc+ca}$
10 replies
+3 w
soryn
Apr 25, 2021
SomeonesPenguin
23 minutes ago
Solve this Equation
NotAHuman_   3
N 23 minutes ago by nextgen2000
x³+y³=253(xy+1)
3 replies
NotAHuman_
Feb 16, 2025
nextgen2000
23 minutes ago
Inequality
stergiu   11
N 32 minutes ago by ektorasmiliotis
Source: Greek olympiad - 2010, senior , today, problem - 2
If $ x,y$ are positive real numbers with sum $ 2a$, prove that :


$ x^3y^3(x^2+y^2)^2 \leq 4a^{10}$

When does equality hold ?

Babis
11 replies
stergiu
Feb 27, 2010
ektorasmiliotis
32 minutes ago
inequality
Butterfly   3
N 44 minutes ago by GreekIdiot

Let $a,b,c>0$ and $a+b+c=ab+bc+ca$. Prove $$\frac{a}{a^3+b^2+1}+\frac{b}{b^3+c^2+1}+\frac{c}{c^3+a^2+1}\le 1.$$
3 replies
Butterfly
Today at 7:36 AM
GreekIdiot
44 minutes ago
Involution Vortex
shanelin-sigma   0
44 minutes ago
Source: 3rd KYAC P2
Given a positive real number $c$. Determine all function $f:\mathbb{R}^+\to\mathbb{R}^+$ such that $$xf(f(y))f(x+y)=c\cdot f(f(x)+f(y))$$for all positive real numbers $x,y$

Prospered by shanelin-sigma
0 replies
shanelin-sigma
44 minutes ago
0 replies
phi(a-b) | f(a)-f(b)
GreenTea2593   3
N an hour ago by lolsamo
Source: KTOM February 2025 Essay Problem 3
Determine all functions $f:\mathbb{N} \to \mathbb{N}$ such that $\varphi(a-b)\mid f(a)-f(b)$ for every coprime positive integers $a>b$.
3 replies
GreenTea2593
Yesterday at 5:00 PM
lolsamo
an hour ago
JBMO Shortlist 2020 C1
Lukaluce   10
N an hour ago by Daniyalimangaziev
Source: JBMO Shortlist 2020
Alice and Bob play the following game: starting with the number $2$ written on a blackboard, each player in turn changes the current number $n$ to a number $n + p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greater than $\underbrace{22...2}_{2020}$. Assuming perfect play, who will win the game.
10 replies
Lukaluce
Jul 4, 2021
Daniyalimangaziev
an hour ago
Inspired by SXTX 2025
sqing   0
an hour ago
Source: Own
Let $ a,b,c>0 $ and $ a+2b+3c=3\sqrt{2}. $ Prove that$$ \sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 3\sqrt{6}$$
0 replies
1 viewing
sqing
an hour ago
0 replies
Easy inequality
Lonesan   2
N an hour ago by KhuongTrang


For $x, y, z$ positive real numbers:


$ \sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x} \ge \sqrt{4(x+y+z)+\frac{6(xy+yz+zx)}{x+y+z}} $


Lorian Saceanu
January2021
2 replies
Lonesan
Jan 31, 2021
KhuongTrang
an hour ago
inequalities
Juno_34   4
N an hour ago by Juno_34
given x,y postive real number , prove that $$\frac{1}{(x+1)^2}+\frac{1}{(y+1)^2} \geqslant \frac{1}{xy+1} $$
4 replies
Juno_34
2 hours ago
Juno_34
an hour ago
A distance on the Euclidean plane
Photaesthesia   5
N an hour ago by R9182
Source: 2025 China Mathematical Olympiad Day 2 Problem 4
The fractional distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is defined as \[ \sqrt{ \left\| x_1 - x_2 \right\|^2 + \left\| y_1 - y_2 \right\|^2},\]where $\left\| x \right\|$ denotes the distance between $x$ and its nearest integer. Find the largest real $r$ such that there exists four points on the plane whose pairwise fractional distance are all at least $r$.
5 replies
Photaesthesia
Nov 28, 2024
R9182
an hour ago
Property of Fermat points related to Neuberg cubic
TelvCohl   2
N Feb 11, 2025 by MuLambda
Source: Own
Given a $ \triangle ABC $ and a point $ P $ lying on the Neuberg cubic of $ \triangle ABC. $ Let $ S $ be the 2nd / 1st Fermat point of $ \triangle ABC $ and $ T, $ $ T_a, $ $ T_b, $ $ T_c $ be the 1st / 2nd Fermat point of $ \triangle ABC, $ $ \triangle BPC, $ $ \triangle CPA, $ $ \triangle APB, $ respectively. Prove that

(a) $ T, $ $ T_a, $ $ T_b, $ $ T_c $ are concyclic.

(b) $ AT_a, $ $ BT_b, $ $ CT_c, $ $ PS $ are concurrent.
2 replies
TelvCohl
Nov 17, 2016
MuLambda
Feb 11, 2025
Property of Fermat points related to Neuberg cubic
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G H BBookmark kLocked kLocked NReply
Source: Own
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TelvCohl
2310 posts
#1 • 7 Y
Y by baopbc, mineiraojose, enhanced, amar_04, Paramizo_Dicrominique, Adventure10, Mango247
Given a $ \triangle ABC $ and a point $ P $ lying on the Neuberg cubic of $ \triangle ABC. $ Let $ S $ be the 2nd / 1st Fermat point of $ \triangle ABC $ and $ T, $ $ T_a, $ $ T_b, $ $ T_c $ be the 1st / 2nd Fermat point of $ \triangle ABC, $ $ \triangle BPC, $ $ \triangle CPA, $ $ \triangle APB, $ respectively. Prove that

(a) $ T, $ $ T_a, $ $ T_b, $ $ T_c $ are concyclic.

(b) $ AT_a, $ $ BT_b, $ $ CT_c, $ $ PS $ are concurrent.
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WizardMath
2487 posts
#2 • 2 Y
Y by Adventure10, Mango247
Beautiful property!
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MuLambda
1 post
#4
Y by
Does anyone here have a synthetic proof of this property please? Or references to published papers?
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