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
Easy Number Theory
VicKmath7   3
N 2 minutes ago by Congruence
Source: Archimedes Junior 2009
If the number $K = \frac{9n^2+31}{n^2+7}$ is integer, find the possible values of $n \in Z$.
3 replies
VicKmath7
Mar 17, 2020
Congruence
2 minutes ago
J
H
Help me to understand 8
youochange   2
N 8 minutes ago by sadat465
∆$ABC$ is a triangle, $M$,$N$ are the midpoints of $AB$, $AC$. $H$ be the orthocenter of $(ABC)$. $NH\cap (ABC)={Q,T} $and $MH \cap (ABC)={P,S}$. Prove that $AHBS$ and $CHAT$ are parallelograms.
2 replies
youochange
Feb 6, 2025
sadat465
8 minutes ago
J
H
inequality ( 4 var
SunnyEvan   9
N 10 minutes ago by SunnyEvan
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{76}{25} \geq \frac{44}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
9 replies
SunnyEvan
Apr 4, 2025
SunnyEvan
10 minutes ago
J
H
FIGO is perpendicular
USJL   6
N 15 minutes ago by WLOGQED1729
Source: 2018 Taiwan TST Round 3
Let $I,G,O$ be the incenter, centroid and the circumcenter of triangle $ABC$, respectively. Let $X,Y,Z$ be on the rays $BC, CA, AB$ respectively so that $BX=CY=AZ$. Let $F$ be the centroid of $XYZ$.

Show that $FG$ is perpendicular to $IO$.
6 replies
1 viewing
USJL
Apr 2, 2020
WLOGQED1729
15 minutes ago
J
No more topics!
H
J
H
Finding a cyclic quadrilateral
wangsacl   9
N Aug 10, 2009 by stergiu
Source: Indonesian MO (INAMO) 2009, Day 2, Problem 8
Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$.
(a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$.
(b) Show that $ A_1KML$ is a cyclic quadrilateral.
9 replies
wangsacl
Aug 8, 2009
stergiu
Aug 10, 2009
J
Finding a cyclic quadrilateral
G H J
Reveal topic tags
geometry
trigonometry
circumcircle
angle bisector
perpendicular bisector
geometry unsolved
L
G H BBookmark kLocked kLocked NReply
Source: Indonesian MO (INAMO) 2009, Day 2, Problem 8
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
wangsacl
182 posts
wangsacl
#1 h  Aug 8, 2009, 9:04 AM • 2 Y
Y by Adventure10, Mango247
Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$.
(a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$.
(b) Show that $ A_1KML$ is a cyclic quadrilateral.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
plane geometry
467 posts
plane geometry
#2 h  Aug 8, 2009, 12:07 PM • 1 Y
Y by Adventure10
according to http://www.mathlinks.ro/viewtopic.php?p=1525553#1525553 we have $ AK\bot BK$ $ AK\bot CL$
Notice ML=MK=MD=c-b => ML//BA
done
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
wangsacl
182 posts
wangsacl
#3 h  Aug 8, 2009, 1:32 PM • 2 Y
Y by Adventure10, Mango247
How do you get $ MK=ML=MD$?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
babylon
15 posts
babylon
#4 h  Aug 8, 2009, 1:56 PM • 2 Y
Y by Adventure10, Mango247
$ MK||AC$ $ \Rightarrow$ $ \angle CED=\angle MKD$ and $ \angle CDE=\angle MDK$ so $ MD=MK$

$ ML||AB$ $ \Rightarrow$ $ \triangle MLD\sim \triangle BDF$ $ \Leftrightarrow$ $ MD=ML$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
plane geometry
467 posts
plane geometry
#5 h  Aug 8, 2009, 2:21 PM • 2 Y
Y by Adventure10, Mango247
wangsacl wrote:
How do you get $ MK = ML = MD$?
AC'=AC => ML=c-b
AB=AB' => MK=c-b
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
plane geometry
467 posts
plane geometry
#6 h  Aug 8, 2009, 2:31 PM • 2 Y
Y by Adventure10, Mango247
babylon wrote:
$ MK||AC$ $ \Rightarrow$ $ \angle CED = \angle MKD$ and $ \angle CDE = \angle MDK$ so $ MD = MK$

$ ML||AB$ $ \Rightarrow$ $ \triangle MLD\sim \triangle BDF$ $ \Leftrightarrow$ $ MD = ML$
No, in this step we have not proved AB//ML because we want use ML=MK=c-b to prove AB//ML MK//AC
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
wangsacl
182 posts
wangsacl
#7 h  Aug 9, 2009, 6:01 AM • 2 Y
Y by Adventure10, Mango247
Wow, nice.

I use trigonometry to prove $ MK=ML$ (which is quite long and complicated)...
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jayme
9775 posts
jayme
#8 h  Aug 9, 2009, 10:36 AM • 2 Y
Y by Adventure10, Mango247
Dear Mathlinkers,
1. for (a) see : http://perso.orange.fr/jl.ayme/ vol. 4, An unlikely concurrence p. 5-6
2. for (b) see : http://perso.orange.fr/jl.ayme/ vol. 1, Feuerbach's theorem p. 3
this is the "Calabre's circle".
Sincerely
Jean-Louis
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jgnr
1343 posts
jgnr
#9 h  Aug 10, 2009, 11:17 AM • 2 Y
Y by Adventure10, Mango247
Another proof of b
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
stergiu
1648 posts
stergiu
#10 h  Aug 10, 2009, 3:45 PM • 2 Y
Y by Adventure10, Mango247
An easy hint for b):

We have

a) $ \angle MLK = \angle A / 2$, since $ LM // AB$

b) $ \angle MA_1K = \angle BA_1K = \angle BAK = \angle A /2$, since $ BKA_1A$ is insribed($ \angle AA_1B = 90 = \angle AKB$).

Hence $ \angle MLK = \angle MA_1K$ .

Babis
Z K Y
N Quick Reply
G
H
=
a