Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

We have your learning goals covered with Spring and Summer courses available. Enroll today!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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 Self-Paced

Prealgebra 1
Sunday, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
Sunday, Apr 13 - Aug 10
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Tuesday, Mar 25 - Jul 8
Sunday, Apr 13 - Aug 10
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21


Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Sunday, Mar 16 - Jun 8
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Monday, Mar 17 - Jun 9
Thursday, Apr 17 - Jul 3
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Sunday, Mar 2 - Jun 22
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
Wednesday, Apr 23 - Oct 1
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
Monday, Apr 21 - Oct 13
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Sunday, Mar 23 - Aug 3
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Sunday, Mar 16 - Aug 24
Wednesday, Apr 9 - Sep 3
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21
Tuesday, Jun 10 - Aug 26

Calculus
Sunday, Mar 30 - Oct 5
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Sunday, Mar 23 - Jun 15
Wednesday, Apr 16 - Jul 2
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!

MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Monday, Mar 24 - Jun 16
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Sunday, Mar 30 - Jun 22
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Tuesday, Mar 25 - Sep 2
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Mar 2, 2025
0 replies
J
H
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
J
H
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
J
H
IMO ShortList 1998, algebra problem 3
orl   69
N 3 minutes ago by Marcus_Zhang
Source: IMO ShortList 1998, algebra problem 3
Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that


\[ \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}. \]
69 replies
+1 w
orl
Oct 22, 2004
Marcus_Zhang
3 minutes ago
J
H
Mathhhhh
mathbetter   5
N 9 minutes ago by mathbetter
Three turtles are crawling along a straight road heading in the same
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?
5 replies
mathbetter
Yesterday at 11:21 AM
mathbetter
9 minutes ago
J
H
IMO ShortList 2001, algebra problem 6
orl   137
N 26 minutes ago by Levieee
Source: IMO ShortList 2001, algebra problem 6
Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]
137 replies
1 viewing
orl
Sep 30, 2004
Levieee
26 minutes ago
J
H
Checkerboard
Ecrin_eren   2
N 39 minutes ago by Thorbeam
On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)
2 replies
Ecrin_eren
Today at 5:20 AM
Thorbeam
39 minutes ago
J
H
9 Three concurrent chords
v_Enhance   1
N 42 minutes ago by YaoAOPS
Three distinct circles $\Omega_1$, $\Omega_2$, $\Omega_3$ cut three common chords concurrent at $X$. Consider two distinct circles $\Gamma_1$, $\Gamma_2$ which are internally tangent to all $\Omega_i$. Determine, with proof, which of the following two statements is true.

(1) $X$ is the insimilicenter of $\Gamma_1$ and $\Gamma_2$
(2) $X$ is the exsimilicenter of $\Gamma_1$ and $\Gamma_2$.
1 reply
+2 w
v_Enhance
an hour ago
YaoAOPS
42 minutes ago
J
H
Simple vector geometry existence
AndreiVila   3
N an hour ago by Ianis
Source: Romanian District Olympiad 2025 9.1
Let $ABCD$ be a parallelogram of center $O$. Prove that for any point $M\in (AB)$, there exist unique points $N\in (OC)$ and $P\in (OD)$ such that $O$ is the center of mass of $\triangle MNP$.
3 replies
AndreiVila
Mar 8, 2025
Ianis
an hour ago
J
H
BD tangent to (MDE) , rhombus ABCD with <DCB=60^o
parmenides51   1
N 2 hours ago by vanstraelen
Source: 2021 Germany R4 10.6 https://artofproblemsolving.com/community/c3208025_
Let a rhombus $ABCD$ with $|\angle DCB| = 60^o$ be given . On the extension of the segment $\overline{CD}$ beyond $D$, a point $E$ is chosen arbitrarily. Let the line through $E$ and $A$ intersect the line $BC$ at the point $F$. Let $M$ be the intersection of the lines $BE$ and $DF$. Prove that the line $BD$ is tangent to the circumcircle of the triangle $MDE$.
1 reply
parmenides51
Oct 6, 2024
vanstraelen
2 hours ago
J
H
Geometry Problem #42
vankhea   2
N 2 hours ago by kaede_Arcadia
Source: Van Khea
Let $P$ be any point. Let $D, E, F$ be projection point from $P$ to $BC, CA, AB$. Circumcircle $(ABC)$ cuts circumcircle $(AEF), (BFD), (CDE)$ at $A_1, B_1, C_1$. Let $A_2, B_2, C_2$ be antipode of $A_1, B_1, C_1$ wrt $(AEF), (BFD), (CDE)$. Prove that $A_2, B_2, C_2, P$ are cyclic.
2 replies
vankhea
Sep 6, 2023
kaede_Arcadia
2 hours ago
J
H
divisibility
srnjbr   3
N 3 hours ago by srnjbr
Find all natural numbers n such that there exists a natural number l such that for every m members of the natural numbers the number m+m^2+...m^l is divisible by n.
3 replies
srnjbr
5 hours ago
srnjbr
3 hours ago
J
H
Very easy inequality
pggp   5
N 3 hours ago by ionbursuc
Source: Polish Junior MO Second Round 2019
Let $x$, $y$ be real numbers, such that $x^2 + x \leq y$. Prove that $y^2 + y \geq x$.
5 replies
pggp
Oct 26, 2020
ionbursuc
3 hours ago
J
H
Solve in gaussian integers
CHESSR1DER   0
3 hours ago
Solve in gaussian integers.
$ \sin\left(\ln\left(x^{x^{x^2}}\right)\right) = x^4 $
0 replies
1 viewing
CHESSR1DER
3 hours ago
0 replies
J
H
Inequality and function
srnjbr   4
N 3 hours ago by srnjbr
Find all f:R--R such that for all x,y, yf(x)+f(y)>=f(xy)
4 replies
srnjbr
5 hours ago
srnjbr
3 hours ago
J
H
Problem 4
blug   3
N 3 hours ago by sunken rock
Source: Polish Junior Math Olympiad Finals 2025
In a rhombus $ABCD$, angle $\angle ABC=100^{\circ}$. Point $P$ lies on $CD$ such that $\angle PBC=20^{\circ}$. Line parallel to $AD$ passing trough $P$ intersects $AC$ at $Q$. Prove that $BP=AQ$.
3 replies
blug
Mar 15, 2025
sunken rock
3 hours ago
J
H
CMI Entrance 19#6
bubu_2001   5
N 4 hours ago by quasar_lord
$(a)$ Compute -
\begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*}$(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $

$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.

$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$
5 replies
bubu_2001
Nov 1, 2019
quasar_lord
4 hours ago
J
H
J
H
Interesting inequality
sqing   1
N Mar 18, 2025 by sqing
Source: Own
Let $ a,b\geq 2 . $ Prove that
$$ (a^2-1)(b^2-1) (1-ab)+\frac{27}{8}a^2b^2\leq 27$$$$ (a^2-1)(b^2-1)(1-a^2b^2 )+\frac{81}{4}a^2b^2 \leq 189$$$$ (a^2-1)(b^2-1)(1-a^2b^2 )+ 162 ab \leq 513$$$$ (a^2-1)(b^2-1) (1-a^2b^2 )+21 a^2b^2\leq \frac{3219}{16}$$$$ (a^2-1)(b^2-1) (1-ab)+\frac{27}{8}a^2b^2\leq\frac{415+61\sqrt{61}}{18}$$
1 reply
sqing
Mar 18, 2025
sqing
Mar 18, 2025
J
Interesting inequality
G H J
Reveal topic tags
inequalities proposed
inequalities
L
G H BBookmark kLocked kLocked NReply
Source: Own
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41157 posts
sqing
#1 h  Mar 18, 2025, 7:50 AM
Y by
Let $ a,b\geq 2 . $ Prove that
$$ (a^2-1)(b^2-1) (1-ab)+\frac{27}{8}a^2b^2\leq 27$$$$ (a^2-1)(b^2-1)(1-a^2b^2 )+\frac{81}{4}a^2b^2 \leq 189$$$$ (a^2-1)(b^2-1)(1-a^2b^2 )+ 162 ab \leq 513$$$$ (a^2-1)(b^2-1) (1-a^2b^2 )+21 a^2b^2\leq \frac{3219}{16}$$$$ (a^2-1)(b^2-1) (1-ab)+\frac{27}{8}a^2b^2\leq\frac{415+61\sqrt{61}}{18}$$
This post has been edited 1 time. Last edited by sqing, Mar 18, 2025, 7:51 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41157 posts
sqing
#2 h  Mar 18, 2025, 8:45 AM
Y by
Let $ a,b\geq 2 . $ Prove that
$$(a^2-1)(b^2-1) -2ab\sqrt{ab}\geq-7$$$$(a^2-1)(b^2-1) -3ab\sqrt{ab}\geq -\frac{93}{4}$$$$(a^3-1)(b^3-1) -42ab\geq-119$$$$(a^3-1)(b^3-1) -43ab \geq -7-\frac{172}{3}\sqrt{\frac{86}{21}}$$$$ (a^3-1)(b^3-1) -14a b\sqrt{a b}\geq -63$$$$(a^3-1)(b^3-1) -15a b\sqrt{a b}\geq-\frac{499}{7}$$
Z K Y
N Quick Reply
G
H
=
a