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

Stay ahead of learning milestones! Enroll in a class over the summer!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/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, 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
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
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
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
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
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
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
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
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
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
Tuesday, Jun 10 - Aug 26

Calculus
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
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
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
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
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
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
Apr 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
2 var inquality
sqing   0
13 minutes ago
Source: Own
Let $ a,b\geq 0 $ and $ a+ b+ab+a^2+b^2=5. $ Prove that
$$ (9+\sqrt{21} ) (a+b-2)(5- 2ab) \ge 10(a-1)(b-1)(a-b)$$$$ (9+\sqrt{21} ) (a+b-2)(3- ab) \ge6(a-1)(b-1)(a-b)$$
0 replies
1 viewing
sqing
13 minutes ago
0 replies
J
H
2 var inquality
sqing   1
N 26 minutes ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a+ b+2ab=4 . $ Prove that
$$ 3(a+b-2)(2 - ab) \ge (a-1)(b-1)(a-b)$$$$ 9 (a+b-2)(3 - 2ab) \ge 2\sqrt 5(a-1)(b-1)(a-b)$$$$9(a+b-2)(6 - 5ab) \ge2\sqrt {14} (a-1)(b-1)(a-b)$$
1 reply
1 viewing
sqing
an hour ago
sqing
26 minutes ago
J
H
Inequalities
hn111009   1
N 44 minutes ago by hn111009
Source: Maybe anywhere?
Let $a,b,c>0;r,s\in\mathbb{R}$ satisfied $a+b+c=1.$ Find minimum and maximum of $$P=a^rb^s+b^rc^s+c^ra^s.$$
1 reply
hn111009
Apr 13, 2025
hn111009
44 minutes ago
J
H
Function equation
luci1337   1
N an hour ago by jasperE3
find all function $f:R \rightarrow R$ such that:
$2f(x)f(x+y)-f(x^2)=\frac{x}{2}(f(2x)+f(f(y)))$ with all $x,y$ is real number
1 reply
luci1337
Yesterday at 3:01 PM
jasperE3
an hour ago
J
H
Integer-Valued FE comes again
lminsl   204
N 2 hours ago by Sedro
Source: IMO 2019 Problem 1
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$Proposed by Liam Baker, South Africa
204 replies
lminsl
Jul 16, 2019
Sedro
2 hours ago
J
H
Quadratic system
juckter   31
N 2 hours ago by blueprimes
Source: Mexico National Olympiad 2011 Problem 3
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:

\[a_1^2 + a_1 - 1 = a_2\]\[ a_2^2 + a_2 - 1 = a_3\]\[\hspace*{3.3em} \vdots \]\[a_{n}^2 + a_n - 1 = a_1\]
31 replies
juckter
Jun 22, 2014
blueprimes
2 hours ago
J
H
The old one is gone.
EeEeRUT   5
N 2 hours ago by Thelink_20
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
5 replies
1 viewing
EeEeRUT
Apr 16, 2025
Thelink_20
2 hours ago
J
H
Another factorisation problem
kjhgyuio   2
N 3 hours ago by kjhgyuio
........
2 replies
kjhgyuio
3 hours ago
kjhgyuio
3 hours ago
J
H
Beautiful geometry
m4thbl3nd3r   1
N 3 hours ago by m4thbl3nd3r
Let $\omega$ be the circumcircle of triangle $ABC$, $M$ is the midpoint of $BC$ and $E$ be the second intersection of $AM$ and $\omega$. Tangent line of $\omega$ at $E$ intersects $BC$ at $P$, let $PKL$ be a transversal of $\omega$ and $X,Y$ be intersections of $AK,AL$ with $BC$. Let $PF$ be a tangent line of $\omega$. Prove that $LYFP$ is cyclic
1 reply
m4thbl3nd3r
Yesterday at 4:41 PM
m4thbl3nd3r
3 hours ago
J
H
integers inequality
azzam2912   0
3 hours ago

3. Let $a, b, c, d, x, y$ be positive integers that satisfy the inequality
\[ \frac{a}{b} < \frac{x}{y} < \frac{c}{d} \]with $bc - ad = 5$. If $b + d = 2025$, determine the minimum value of $y$.


0 replies
azzam2912
3 hours ago
0 replies
J
H
MONT pg 31 example 1.10.40
Jaxman8   0
5 hours ago
Can somebody explain why it works.
0 replies
Jaxman8
5 hours ago
0 replies
J
H
NEPAL TST 2025 DAY 2
Tony_stark0094   9
N 5 hours ago by hectorleo123
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.

If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
9 replies
Tony_stark0094
Apr 12, 2025
hectorleo123
5 hours ago
J
H
lines CV, BU intersect on the circumcircle of ABC
parmenides51   4
N 5 hours ago by ihategeo_1969
Source: 2019 Geo Mock - Olympiad by Tovi Wen #4 https://artofproblemsolving.com/community/c594864h1787237p11805928
Let $ABC$ be a triangle whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D, E, F$, respectively. Let $M$ and $N$ be the midpoints of $\overline{DE}$ and $\overline{DF}$, respectively. Suppose that points $U$ and $V$ lie on $\overline{MN}$ so that $BU = NU$ and $CV = MV$. Prove that lines $\overline{CV}$ and $\overline{BU}$ intersect on the circumcircle of $\triangle ABC$.
4 replies
parmenides51
Nov 26, 2023
ihategeo_1969
5 hours ago
J
H
one cyclic formed by two cyclic
CrazyInMath   33
N 5 hours ago by breloje17fr
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
33 replies
CrazyInMath
Apr 13, 2025
breloje17fr
5 hours ago
J
H
J
H
locus of AP x CP + BPx DP = 1 where ABCD square of sidelength 1
parmenides51   2
N Jul 17, 2020 by WolfusA
Source: Switzerland - Swiss MO 2008 p5
Let $ABCD$ be a square with side length $1$.
Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.
2 replies
parmenides51
Jul 17, 2020
WolfusA
Jul 17, 2020
J
locus of AP x CP + BPx DP = 1 where ABCD square of sidelength 1
G H J
Reveal topic tags
geometry
Locus
square
L
G H BBookmark kLocked kLocked NReply
Source: Switzerland - Swiss MO 2008 p5
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
parmenides51
30629 posts
parmenides51
#1 h  Jul 17, 2020, 6:28 AM
Y by
Let $ABCD$ be a square with side length $1$.
Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Gryphos
1702 posts
Gryphos
#2 h  Jul 17, 2020, 8:33 AM • 2 Y
Y by Mango247, Mango247
Solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
WolfusA
1900 posts
WolfusA
#3 h  Jul 17, 2020, 9:30 AM
Y by
Let's keep it classic. We will prove the following theorem:
In a square $ABCD$ with side length $a$ holds $$AP\cdot CP + BP\cdot DP = a^2\iff \left(P\in \overline{AC}\vee P\in \overline{BD}\right)$$proof
comment to above
This post has been edited 1 time. Last edited by WolfusA, Jul 17, 2020, 9:32 AM
Reason: adding comment
Z K Y
N Quick Reply
G
H
=
a