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
Basic geometry
AlexCenteno2007   0
30 minutes ago
Let $ABC$ be an equilateral triangle.M and N are the midpoints of $AB$ and $BC$ respectively. Externally to the triangle $ABC$ an isosceles right triangle APC is constructed, with the angle $APC =90°$.If point $I$ is the intersection of $AN$ and $MP$, show that $CI$ is a bisector of the angle $ACM$
0 replies
AlexCenteno2007
30 minutes ago
0 replies
J
H
Geometry Basic
AlexCenteno2007   1
N 38 minutes ago by mickeymouse7133
Let $ABC$ be an isosceles triangle such that $AC=BC$. Let $P$ be a dot on the $AC$ side.
The tangent to the circumcircle of $ABP$ at point $P$ intersects the circumcircle of $BCP$ at $D$. Prove that CD$ \parallel$AB
1 reply
AlexCenteno2007
40 minutes ago
mickeymouse7133
38 minutes ago
J
H
BrUMO 2025 Team Round Problem 2
lpieleanu   1
N 42 minutes ago by MathCosine
Two squares, $ABCD$ and $AEFG,$ have equal side length $x.$ They intersect at $A$ and $O.$ Given that $CO=2$ and $OA=2\sqrt{2},$ what is $x$?
1 reply
lpieleanu
2 hours ago
MathCosine
42 minutes ago
J
H
\phi (n) related 2024 TMC AIME Mock #7
parmenides51   1
N 42 minutes ago by maromex
Let $\phi (n)$ denote the number of positive integers less than$ n$ that are relatively prime to $n$. Let $S$ denote the set of positive integers $n$ such that when $\frac{n}{\phi (n)}$ is expressed as a simplified fraction, the denominator is a power of $2$. Find the smallest prime number $p$ that satisfies the following:
$\bullet$ $p -1$ is not divisible by any square of a positive integer greater than $1$,
$\bullet$ No element in $S$ is divisible by $p$.
1 reply
parmenides51
Saturday at 8:08 PM
maromex
42 minutes ago
J
H
Geometric inequality with Fermat point
Assassino9931   1
N 2 hours ago by Circumcircle
Source: Balkan MO Shortlist 2024 G2
Let $ABC$ be an acute triangle and let $P$ be an interior point for it such that $\angle APB = \angle BPC = \angle CPA$. Prove that
$$ \frac{PA^2 + PB^2 + PC^2}{2S} + \frac{4}{\sqrt{3}} \leq \frac{1}{\sin \alpha} + \frac{1}{\sin \beta} + \frac{1}{\sin \gamma}. $$When does equality hold?
1 reply
Assassino9931
3 hours ago
Circumcircle
2 hours ago
J
H
Involved conditional geo
Assassino9931   1
N 2 hours ago by hukilau17
Source: Balkan MO 2024 Shortlist G4
Let $ABC$ be an acute-angled triangle with $AB < AC$, orthocenter $H$, circumcircle $\Gamma$ and circumcentre $O$. Let $M$ be the midpoint of $BC$ and let $D$ be a point such that $ADOH$ is a parallellogram. Suppose that there exists a point $X$ on $\Gamma$ and on the opposite side of $DH$ to $A$ such that $\angle DXH + \angle DHA = 90^{\circ}$. Let $Y$ be the midpoint of $OX$. Prove that if $MY = OA$, then $OA = 2OH$.
1 reply
Assassino9931
2 hours ago
hukilau17
2 hours ago
J
H
Inversion exercise
Assassino9931   2
N 2 hours ago by awesomeming327.
Source: Balkan MO Shortlist 2024 G5
Let $ABC$ be an acute scalene triangle $ABC$, $D$ be the orthogonal projection of $A$ on $BC$, $M$ and $N$ are the midpoints of $AB$ and $AC$ respectively. Let $P$ and $Q$ are points on the minor arcs $\widehat{AB}$ and $\widehat{AC}$ of the circumcircle of triangle $ABC$ respectively such that $PQ \parallel BC$. Show that the circumcircles of triangles $DPQ$ and $DMN$ are tangent if and only if $M$ lies on $PQ$.
2 replies
Assassino9931
2 hours ago
awesomeming327.
2 hours ago
J
H
One more problem defined only with lines
Assassino9931   0
2 hours ago
Source: Balkan MO 2024 Shortlist G6
Let $ABC$ be a triangle and the points $K$ and $L$ on $AB$, $M$ and $N$ on $BC$, and $P$ and $Q$ on $AC$ be such that $AK = LB < \frac{1}{2}AB, BM = NC < \frac{1}{2}BC$ and $CP = QA < \frac{1}{2}AC$. The intersections of $KN$ with $MQ$ and $LP$ are $R$ and $T$ respectively, and the intersections of $NP$ with $LM$ and $KQ$ are $D$ and $E$, respectively. Prove that the lines $DR, BE$ and $CT$ are concurrent.
0 replies
Assassino9931
2 hours ago
0 replies
J
H
Fixed point in a small configuration
Assassino9931   0
2 hours ago
Source: Balkan MO Shortlist 2024 G3
Let $A, B, C, D$ be fixed points on this order on a line. Let $\omega$ be a variable circle through $C$ and $D$ and suppose it meets the perpendicular bisector of $CD$ at the points $X$ and $Y$. Let $Z$ and $T$ be the other points of intersection of $AX$ and $BY$ with $\omega$. Prove that $ZT$ passes through a fixed point independent of $\omega$.
0 replies
Assassino9931
2 hours ago
0 replies
J
H
Sum of divisors
DinDean   1
N 3 hours ago by Tintarn
Does there exist $M>0$, such that $\forall m>M$, there exists an integer $n$ satisfying $\sigma(n)=m$?
$\sigma(n)=$ the sum of all positive divisors of $n$.
1 reply
DinDean
Apr 18, 2025
Tintarn
3 hours ago
J
H
Interesting polygon game
Assassino9931   0
3 hours ago
Source: Balkan MO Shortlist 2024 C5
Let $n\geq 3$ be an integer. Alice and Bob play the following game on the vertices of a regular $n$-gon. Alice places her token on a vertex of the n-gon. Afterwards Bob places his token on another vertex of the n-gon. Then, with Alice playing first, they move their tokens alternately as follows for $2n$ rounds: In Alice’s turn on the $k$-th round, she moves her token $k$ positions clockwise or anticlockwise. In Bob’s turn on the $k$-th round, he moves his token $1$ position clockwise or anticlockwise. If at the end of any person’s turn the two tokens are on the same vertex, then Alice wins the game, otherwise Bob wins. Decide for each value of $n$ which player has a winning strategy.
0 replies
Assassino9931
3 hours ago
0 replies
J
H
An equation from the past with different coefficients
Assassino9931   13
N 3 hours ago by grupyorum
Source: Balkan MO Shortlist 2024 N2
Let $n$ be an integer. Prove that $n^4 - 12n^2 + 144$ is not a perfect cube of an integer.
13 replies
Assassino9931
Yesterday at 1:00 PM
grupyorum
3 hours ago
J
H
Euler Totient optimality - why combinatorics?
Assassino9931   0
3 hours ago
Source: Balkan MO Shortlist 2024 C4
Let $k$ be a positive integer. Prove that there exists a positive integer $n$ and distinct primes $p_1,p_2,\ldots,p_k$ such that if $A(n)$ denotes the number of positive integers less than or equal to $n$ and not divisible by any of $p_1,p_2,\ldots,p_k$, then
$$ \left|n\left(1 - \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)\cdots \left(1-\frac{1}{p_k}\right) - A(n)\right| > 2^{k-3} $$
0 replies
Assassino9931
3 hours ago
0 replies
J
H
Abstraction function in combinatorics
Assassino9931   0
3 hours ago
Source: Balkan MO Shortlist 2024 C2
Let $n\geq 2$ be an integer and denote $S = \{1,2,\ldots,n^2\}$. For a function $f: S \to S$ we denote Im $f = \{b\in S: \exists a\in S, f(a) = b\}$, Fix $f = \{x \in S: f(x) = x\}$ and $f^{-1}(k) = \{a\in S: f(a) = k\}$. Find all possible values of $|$Im $f|$ + $|$Fix $f|$ + $\max_{k\in S} |f^{-1}(k)|$.
0 replies
Assassino9931
3 hours ago
0 replies
J
H
J
H
criterion for conguent quadr. (2017-18 Savin Competition 7-9 p17)
parmenides51   0
Jun 21, 2021
Petya invented a creiterion of congruence of quadrangles. He claims that given quadrangles $ABCD$ and $A'B'C'D'$ (not necessarily convex), where the three sides of one are respectively equal to the three sides of the other ($AB=A'B'$, $BC=B'C'$, $CD=C'D'$) and diagonals of one are respectively equal todiagonals of the other ($AC=A'C'$, $BD= B'D'$), then the quadrangles are congruent. Is Petya wrong?
0 replies
parmenides51
Jun 21, 2021
0 replies
J
criterion for conguent quadr. (2017-18 Savin Competition 7-9 p17)
G H J
Reveal topic tags
geometry
congruence
L
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
parmenides51
30650 posts
parmenides51
#1 h  Jun 21, 2021, 9:26 PM • 1 Y
Y by Mango247
Petya invented a creiterion of congruence of quadrangles. He claims that given quadrangles $ABCD$ and $A'B'C'D'$ (not necessarily convex), where the three sides of one are respectively equal to the three sides of the other ($AB=A'B'$, $BC=B'C'$, $CD=C'D'$) and diagonals of one are respectively equal todiagonals of the other ($AC=A'C'$, $BD= B'D'$), then the quadrangles are congruent. Is Petya wrong?
This post has been edited 2 times. Last edited by parmenides51, Jun 21, 2021, 9:27 PM
Z K Y
N Quick Reply
G
H
=
a