Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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 upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/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
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
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, 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
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
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
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
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

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
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
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
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
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
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

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

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
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 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
BMO 2024 SL A4
MuradSafarli   3
N 3 minutes ago by sqing
A4.
Let \(a \geq b \geq c \geq 0\) be real numbers such that \(ab + bc + ca = 3\).
Prove that:
\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]and determine all the cases when the equality occurs.
3 replies
MuradSafarli
Apr 27, 2025
sqing
3 minutes ago
an exponential inequality with two variables
teresafang   5
N 5 minutes ago by teresafang
x and y are positive real numbers.prove that [(x^y)/y]^(1/2)+[(y^x)/x]^(1/2)>=2.
sorry.I’m not good at English.Also I don’t know how to use Letax.
5 replies
teresafang
May 4, 2025
teresafang
5 minutes ago
Inspired by Austria 2025
sqing   6
N 6 minutes ago by sqing
Source: Own
Let $ a,b\geq 0 ,a,b\neq 1$ and $  a^2+b^2=1. $ Prove that$$   (a + b ) \left( \frac{a}{(b -1)^2} + \frac{b}{(a - 1)^2} \right) \geq 12+8\sqrt 2$$
6 replies
sqing
Today at 2:01 AM
sqing
6 minutes ago
Geometry
gggzul   4
N 10 minutes ago by gggzul
In trapezoid $ABCD$ segments $AB$ and $CD$ are parallel. Angle bisectors of $\angle A$ and $\angle C$ meet at $P$. Angle bisectors of $\angle B$ and $\angle D$ meet at $Q$. Prove that $ABPQ$ is cyclic
4 replies
gggzul
Today at 8:22 AM
gggzul
10 minutes ago
Number Theory
fasttrust_12-mn   9
N 12 minutes ago by Shiny_zubat
Source: Pan African Mathematics Olympiad P1
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
9 replies
1 viewing
fasttrust_12-mn
Aug 15, 2024
Shiny_zubat
12 minutes ago
Interesting inequalities
sqing   5
N 16 minutes ago by sqing
Source: Own
Let $ a,b> 0 $ and $  a^2+ab+b^2=a+b   $. Prove that
$$   \frac{a }{2b^2+1}+ \frac{b }{2a^2+1}+ \frac{1}{2ab+1} \geq \frac{21}{17}$$Let $ a,b> 0 $ and $ a^2+ab+b^2=a+b+1   $. Prove that
$$   \frac{a }{2b^2+1}+ \frac{b }{2a^2+1}+ \frac{1}{2ab+1} \geq1$$
5 replies
1 viewing
sqing
May 4, 2025
sqing
16 minutes ago
3-var inequality
sqing   4
N 17 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 ,a+b+c =1. $ Prove that
$$\frac{ab}{2c+1} +\frac{bc}{2a+1} +\frac{ca}{2b+1}+\frac{27}{20} abc\leq \frac{1}{4} $$
4 replies
sqing
May 3, 2025
sqing
17 minutes ago
Incentre-excentre geometry
oVlad   1
N 37 minutes ago by mashumaro
Source: Romania Junior TST 2025 Day 2 P2
Consider a scalene triangle $ABC$ with incentre $I$ and excentres $I_a,I_b,$ and $I_c$, opposite the vertices $A,B,$ and $C$ respectively. The incircle touches $BC,CA,$ and $AB$ at $E,F,$ and $G$ respectively. Prove that the circles $IEI_a,IFI_b,$ and $IGI_c$ have a common point other than $I$.
1 reply
oVlad
2 hours ago
mashumaro
37 minutes ago
IMO Genre Predictions
ohiorizzler1434   53
N 42 minutes ago by GreekIdiot
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
53 replies
1 viewing
ohiorizzler1434
May 3, 2025
GreekIdiot
42 minutes ago
Functional Equation Problem
LeatuyrBertyk   0
an hour ago
Find all function $f:\mathbb{R}\to\mathbb{R}$ such that:
i) $f(x+y)\leq f(x)+f(y),\forall x,y\in\mathbb{R}$;
ii) $\ln 2025\cdot f(x)\leq 2025^x-1,\forall x\in\mathbb{R}$.
0 replies
LeatuyrBertyk
an hour ago
0 replies
Two equal angles
jayme   5
N an hour ago by Captainscrubz
Dear Mathlinkers,

1. ABCD a square
2. I the midpoint of AB
3. 1 the circle center at A passing through B
4. Q the point of intersection of 1 with the segment IC
5. X the foot of the perpendicular to BC from Q
6. Y the point of intersection of 1 with the segment AX
7. M the point of intersection of CY and AB.

Prove : <ACI = <IYM.

Sincerely
Jean-Louis
5 replies
jayme
May 2, 2025
Captainscrubz
an hour ago
Geometry
Lukariman   1
N an hour ago by Lukariman
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that <HDM = 2∠AMP.
1 reply
Lukariman
2 hours ago
Lukariman
an hour ago
5-th powers is a no-go - JBMO Shortlist
WakeUp   7
N an hour ago by Namisgood
Prove that there are are no positive integers $x$ and $y$ such that $x^5+y^5+1=(x+2)^5+(y-3)^5$.

Note
7 replies
WakeUp
Oct 30, 2010
Namisgood
an hour ago
positive integers forming a perfect square
cielblue   5
N an hour ago by Assassino9931
Find all positive integers $n$ such that $2^n-n^2+1$ is a perfect square.
5 replies
cielblue
May 2, 2025
Assassino9931
an hour ago
3 concurrent diagonals
ddziabenko   2
N Jul 30, 2005 by darij grinberg
Source: Austrian MO 2005 round 1
We're given two congruent, equilateral triangles $ABC$ and $PQR$ with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon $A_1A_2A_3A_4A_5A_6$ (labelled counterclockwise). Prove that $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent.
2 replies
ddziabenko
Jun 27, 2005
darij grinberg
Jul 30, 2005
3 concurrent diagonals
G H J
Source: Austrian MO 2005 round 1
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ddziabenko
42 posts
#1 • 2 Y
Y by Adventure10, Mango247
We're given two congruent, equilateral triangles $ABC$ and $PQR$ with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon $A_1A_2A_3A_4A_5A_6$ (labelled counterclockwise). Prove that $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
yetti
2643 posts
#2 • 2 Y
Y by Adventure10, Mango247
Since the corresponding sides $AB \parallel PQ,\ BC \parallel QR,\ CA \parallel RP$ of the equilateral triangles $\triangle ABC \cong \triangle PQR$ are parallel, these 2 triangles are centrally similar, which means that the lines $AP, BQ, CR$ connecting their corresponding vertices concur at their similarity center $H$. Since the 2 equilateral triangles are oppositely congruent, their similarity coefficient is -1. Therefore, for the oriented segments, we have $HA = -HP,\ HB = -HQ,\ HC = -HR$ and consequently, $H$ is the common midpoint of the segments $AP, BQ, CR$

The quadrilateral $AA_2PA_5$ is a parallelogram, because the lines $AA_2 \equiv AB \parallel PQ \equiv QA_5$ and $A_5A \equiv CA \parallel RP \equiv A_2P$ are parallel. Hence, its diagonals $AP, A_2A_5$ cut each other in half at the midpoint $H$ of $AP$.

The quadrilateral $BA_4QA_1$ is a parallelogram, because the lines $BA_4 \equiv BC \parallel QR \equiv QA_1$ and $A_1B \equiv AB \parallel PQ \equiv A_4Q$ are parallel. Hence, its diagonals $BQ, A_1A_4$ cut each other in half at the midpoint $H$ of $BQ$.

The quadrilateral $CA_6RA_3$ is a parallelogram, because the lines $CA_6 \equiv CA \parallel RP \equiv RA_3$ and $A_3C \equiv BC \parallel QR \equiv A_6R$ are parallel. Hence, its diagonals $CR, A_3A_6$ cut each other in half at the midpoint $H$ of $CR$.

As a result, $A_1A_4,\ A_2A_5,\ A_3A_6$ all pass through and concur at the similarity center $H$.
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
darij grinberg
6555 posts
#3 • 2 Y
Y by Adventure10, Mango247
There is an even simpler way of formulating the solution:

Since the triangles ABC and PQR have parallel sides, they are homothetic, and since they are congruent, their homothetic factor is either 1 or -1 (where 1 would actually mean that there is a translation mapping one to the other). The condition that one of the triangles has one vertex pointing up and the other one has the corresponding vertex pointing down specifies that the factor is -1 rather than 1. Thus, there is a homothety with factor -1, i. e. a reflection in a point, which maps the triangle ABC to the triangle PQR. In other words: There exists a point Z such that the triangle PQR is the reflection of the triangle ABC in the point Z. More precisely, the reflection in the point Z maps the points A, B, C to the points P, Q, R.

The points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ were defined as points of intersections of the sides of triangles ABC and PQR, without specifying which point is the intersection of which sides; let's WLOG assume that $A_1=CA\cap QR$, $A_2=AB\cap QR$, $A_3=AB\cap RP$, $A_4=BC\cap RP$, $A_5=BC\cap PQ$ and $A_6=CA\cap PQ$. The reflection in the point Z maps the points A, B, C to the points P, Q, R. Of course, it must then also map the points P, Q, R back to the points A, B, C. Hence, it maps the point $CA\cap QR$ to the point $RP\cap BC$. In other words, our reflection maps the point $A_1$ to the point $A_4$. Hence, the line $A_1A_4$ passes through the center Z of our reflection. Similarly, the lines $A_2A_5$ and $A_3A_6$ also pass through this center Z. Hence, the lines $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent (namely, at the point Z).

Darij
Z K Y
N Quick Reply
G
H
=
a