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
Iran TST P8
TheBarioBario   7
N 5 minutes ago by bin_sherlo
Source: Iranian TST 2022 problem 8
In triangle $ABC$, with $AB<AC$, $I$ is the incenter, $E$ is the intersection of $A$-excircle and $BC$. Point $F$ lies on the external angle bisector of $BAC$ such that $E$ and $F$ lieas on the same side of the line $AI$ and $\angle AIF=\angle AEB$. Point $Q$ lies on $BC$ such that $\angle AIQ=90$. Circle $\omega_b$ is tangent to $FQ$ and $AB$ at $B$, circle $\omega_c$ is tangent to $FQ$ and $AC$ at $C$ and both circles pass through the inside of triangle $ABC$. if $M$ is the Midpoint od the arc $BC$, which does not contain $A$, prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$.

Proposed by Amirmahdi Mohseni
7 replies
TheBarioBario
Apr 2, 2022
bin_sherlo
5 minutes ago
J
H
Parallel lines with incircle
buratinogigle   1
N 25 minutes ago by luutrongphuc
Source: Own, test for the preliminary team of HSGS 2025
Let $ABC$ be a triangle with incircle $(I)$, which touches sides $CA$ and $AB$ at points $E$ and $F$, respectively. Choose points $M$ and $N$ on the line $EF$ such that $BM = BF$ and $CN = CE$. Let $P$ be the intersection of lines $CM$ and $BN$. Define $Q$ and $R$ as the intersections of $PN$ and $PM$ with lines $IC$ and $IB$, respectively. Assume that $J$ is the intersection of $QR$ and $BC$. Prove that $PJ \parallel MN$.
1 reply
buratinogigle
Yesterday at 11:23 AM
luutrongphuc
25 minutes ago
J
H
Inequality with x,y
GeoMorocco   1
N 35 minutes ago by Mathzeus1024
Let $x,y\ge 0$ such that $ 5(x^3+y^3) \leq 16(1+xy)$. Prove that:
$$8+xy\geq 3(x+y) $$
1 reply
GeoMorocco
Apr 20, 2025
Mathzeus1024
35 minutes ago
J
H
Function from the plane to the real numbers
AndreiVila   3
N 39 minutes ago by ItzsleepyXD
Source: Balkan MO Shortlist 2024 G7
Let $f:\pi\rightarrow\mathbb{R}$ be a function from the Euclidean plane to the real numbers such that $$f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$$for any acute triangle $ABC$ with circumcenter $O$, centroid $G$ and orthocenter $H$. Prove that $f$ is constant.
3 replies
AndreiVila
4 hours ago
ItzsleepyXD
39 minutes ago
J
No more topics!
H
J
H
19th kmo #5
lightrhee   3
N Sep 18, 2024 by hanulyeongsam
Source: KMO round 2, problem 5
Let $P$ be a point that lies outside of circle $O$. A line passes through $P$ and meets the circle at $A$ and $B$, and another line passes through $P$ and meets the circle at $C$ and $D$. The point $A$ is between $P$ and $B$, $C$ is between $P$ and $D$. Let the intersection of segment $AD$ and $BC$ be $L$ and construct $E$ on ray $(PA$ so that $BL \cdot PE = DL \cdot PD$.

Show that $M$ is the midpoint of the segment $DE$, where $M$ is the intersection of lines $PL$ and $DE$.
3 replies
lightrhee
Feb 3, 2006
hanulyeongsam
Sep 18, 2024
J
19th kmo #5
G H J
Reveal topic tags
geometry unsolved
geometry
L
G H BBookmark kLocked kLocked NReply
Source: KMO round 2, problem 5
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lightrhee
557 posts
lightrhee
#1 h  Feb 3, 2006, 3:11 AM • 2 Y
Y by Adventure10 and 1 other user
Let $P$ be a point that lies outside of circle $O$. A line passes through $P$ and meets the circle at $A$ and $B$, and another line passes through $P$ and meets the circle at $C$ and $D$. The point $A$ is between $P$ and $B$, $C$ is between $P$ and $D$. Let the intersection of segment $AD$ and $BC$ be $L$ and construct $E$ on ray $(PA$ so that $BL \cdot PE = DL \cdot PD$.

Show that $M$ is the midpoint of the segment $DE$, where $M$ is the intersection of lines $PL$ and $DE$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GHAVR
68 posts
GHAVR
#2 h  Feb 3, 2006, 4:49 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
lightrhee wrote:
This is the fifth problem of the second round of 19th Korean Mathematics Olympiad.
5.
P is a point that is outside of circle O. A line passes through P and meets the circle at A and B, and another line passes through P and meets the circle at C and D. A is between P and B, C is between P and D. Let the intersection of segment AD and BC be L and construct E on ray PA so that $BL \cdot PE = DL \cdot PD$. Show that M is the midpoint of the segment DE, where M is the intersection of line PL and DE.


We have $\frac{BL}{DL}=\frac{PD}{PE} \Rightarrow$
Under the sin theorem we have $\frac{EM}{MD}=\frac{EP sin \angle BPL}{DP sin \angle DPL}=\frac{DL sin \angle BPL}{BL sin \angle DPL}$

$\frac{DL}{sin \angle DPL} * \frac{sin \angle BPL}{BL} = \frac{LP}{sin \angle PDL} * \frac{sin \angle PBL}{LP} = 1$ (as $\angle LDP=\angle PBL$)

Nice problem :)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Sailor
256 posts
Sailor
#3 h  Mar 18, 2006, 1:24 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Since $\frac{BL}{DL}=\frac{PD}{PE}=\frac{AL}{LC}$ $\Longleftrightarrow$ $S_{\triangle{PEL}}=S_{\triangle{PLD}}$ and the conclusion follows.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hanulyeongsam
170 posts
hanulyeongsam
#4 h  Sep 18, 2024, 2:25 AM
Y by
BL:DL=triangle LAB:triangle LCD=x:y
Let H_1,H_4 vertically from M CD,AB
H_2,H_3 vertically from the meeting point of AD and BC to CD,AB

BL:DL=LH_3:LH_2=MH_4:MH_1 MH_4=bx, MH_1=by
BL:DL=PD:PE=> PD=ax,PE=ay
|triangle MEP|=1/2*ay*bx=abxy/2
|triangle MDP|=1/2*ax*by=abxy/2
|MEP|=|MDP| so, MD=ME
Z K Y
N Quick Reply
G
H
=
a