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
Thursday at 11:16 PM
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
Thursday at 11:16 PM
0 replies
k i Peer-to-Peer Programs Forum
jwelsh   157
N Dec 11, 2023 by cw357
Many of our AoPS Community members share their knowledge with their peers in a variety of ways, ranging from creating mock contests to creating real contests to writing handouts to hosting sessions as part of our partnership with schoolhouse.world.

To facilitate students in these efforts, we have created a new Peer-to-Peer Programs forum. With the creation of this forum, we are starting a new process for those of you who want to advertise your efforts. These advertisements and ensuing discussions have been cluttering up some of the forums that were meant for other purposes, so we’re gathering these topics in one place. This also allows students to find new peer-to-peer learning opportunities without having to poke around all the other forums.

To announce your program, or to invite others to work with you on it, here’s what to do:

1) Post a new topic in the Peer-to-Peer Programs forum. This will be the discussion thread for your program.

2) Post a single brief post in this thread that links the discussion thread of your program in the Peer-to-Peer Programs forum.

Please note that we’ll move or delete any future advertisement posts that are outside the Peer-to-Peer Programs forum, as well as any posts in this topic that are not brief announcements of new opportunities. In particular, this topic should not be used to discuss specific programs; those discussions should occur in topics in the Peer-to-Peer Programs forum.

Your post in this thread should have what you're sharing (class, session, tutoring, handout, math or coding game/other program) and a link to the thread in the Peer-to-Peer Programs forum, which should have more information (like where to find what you're sharing).
157 replies
jwelsh
Mar 15, 2021
cw357
Dec 11, 2023
k i C&P posting recs by mods
v_Enhance   0
Jun 12, 2020
The purpose of this post is to lay out a few suggestions about what kind of posts work well for the C&P forum. Except in a few cases these are mostly meant to be "suggestions based on historical trends" rather than firm hard rules; we may eventually replace this with an actual list of firm rules but that requires admin approval :) That said, if you post something in the "discouraged" category, you should not be totally surprised if it gets locked; they are discouraged exactly because past experience shows they tend to go badly.
-----------------------------
1. Program discussion: Allowed
If you have questions about specific camps or programs (e.g. which classes are good at X camp?), these questions fit well here. Many camps/programs have specific sub-forums too but we understand a lot of them are not active.
-----------------------------
2. Results discussion: Allowed
You can make threads about e.g. how you did on contests (including AMC), though on AMC day when there is a lot of discussion. Moderators and administrators may do a lot of thread-merging / forum-wrangling to keep things in one place.
-----------------------------
3. Reposting solutions or questions to past AMC/AIME/USAMO problems: Allowed
This forum contains a post for nearly every problem from AMC8, AMC10, AMC12, AIME, USAJMO, USAMO (and these links give you an index of all these posts). It is always permitted to post a full solution to any problem in its own thread (linked above), regardless of how old the problem is, and even if this solution is similar to one that has already been posted. We encourage this type of posting because it is helpful for the user to explain their solution in full to an audience, and for future users who want to see multiple approaches to a problem or even just the frequency distribution of common approaches. We do ask for some explanation; if you just post "the answer is (B); ez" then you are not adding anything useful.

You are also encouraged to post questions about a specific problem in the specific thread for that problem, or about previous user's solutions. It's almost always better to use the existing thread than to start a new one, to keep all the discussion in one place easily searchable for future visitors.
-----------------------------
4. Advice posts: Allowed, but read below first
You can use this forum to ask for advice about how to prepare for math competitions in general. But you should be aware that this question has been asked many many times. Before making a post, you are encouraged to look at the following:
[list]
[*] Stop looking for the right training: A generic post about advice that keeps getting stickied :)
[*] There is an enormous list of links on the Wiki of books / problems / etc for all levels.
[/list]
When you do post, we really encourage you to be as specific as possible in your question. Tell us about your background, what you've tried already, etc.

Actually, the absolute best way to get a helpful response is to take a few examples of problems that you tried to solve but couldn't, and explain what you tried on them / why you couldn't solve them. Here is a great example of a specific question.
-----------------------------
5. Publicity: use P2P forum instead
See https://artofproblemsolving.com/community/c5h2489297_peertopeer_programs_forum.
Some exceptions have been allowed in the past, but these require approval from administrators. (I am not totally sure what the criteria is. I am not an administrator.)
-----------------------------
6. Mock contests: use Mock Contests forum instead
Mock contests should be posted in the dedicated forum instead:
https://artofproblemsolving.com/community/c594864_aops_mock_contests
-----------------------------
7. AMC procedural questions: suggest to contact the AMC HQ instead
If you have a question like "how do I submit a change of venue form for the AIME" or "why is my name not on the qualifiers list even though I have a 300 index", you would be better off calling or emailing the AMC program to ask, they are the ones who can help you :)
-----------------------------
8. Discussion of random math problems: suggest to use MSM/HSM/HSO instead
If you are discussing a specific math problem that isn't from the AMC/AIME/USAMO, it's better to post these in Middle School Math, High School Math, High School Olympiads instead.
-----------------------------
9. Politics: suggest to use Round Table instead
There are important conversations to be had about things like gender diversity in math contests, etc., for sure. However, from experience we think that C&P is historically not a good place to have these conversations, as they go off the rails very quickly. We encourage you to use the Round Table instead, where it is much more clear that all posts need to be serious.
-----------------------------
10. MAA complaints: discouraged
We don't want to pretend that the MAA is perfect or that we agree with everything they do. However, we chose to discourage this sort of behavior because in practice most of the comments we see are not useful and some are frankly offensive.
[list] [*] If you just want to blow off steam, do it on your blog instead.
[*] When you have criticism, it should be reasoned, well-thought and constructive. What we mean by this is, for example, when the AOIME was announced, there was great outrage about potential cheating. Well, do you really think that this is something the organizers didn't think about too? Simply posting that "people will cheat and steal my USAMOO qualification, the MAA are idiots!" is not helpful as it is not bringing any new information to the table.
[*] Even if you do have reasoned, well-thought, constructive criticism, we think it is actually better to email it the MAA instead, rather than post it here. Experience shows that even polite, well-meaning suggestions posted in C&P are often derailed by less mature users who insist on complaining about everything.
[/list]
-----------------------------
11. Memes and joke posts: discouraged
It's fine to make jokes or lighthearted posts every so often. But it should be done with discretion. Ideally, jokes should be done within a longer post that has other content. For example, in my response to one user's question about olympiad combinatorics, I used a silly picture of Sogiita Gunha, but it was done within a context of a much longer post where it was meant to actually make a point.

On the other hand, there are many threads which consist largely of posts whose only content is an attached meme with the word "MAA" in it. When done in excess like this, the jokes reflect poorly on the community, so we explicitly discourage them.
-----------------------------
12. Questions that no one can answer: discouraged
Examples of this: "will MIT ask for AOIME scores?", "what will the AIME 2021 cutoffs be (asked in 2020)", etc. Basically, if you ask a question on this forum, it's better if the question is something that a user can plausibly answer :)
-----------------------------
13. Blind speculation: discouraged
Along these lines, if you do see a question that you don't have an answer to, we discourage "blindly guessing" as it leads to spreading of baseless rumors. For example, if you see some user posting "why are there fewer qualifiers than usual this year?", you should not reply "the MAA must have been worried about online cheating so they took fewer people!!". Was sich überhaupt sagen lässt, lässt sich klar sagen; und wovon man nicht reden kann, darüber muss man schweigen.
-----------------------------
14. Discussion of cheating: strongly discouraged
If you have evidence or reasonable suspicion of cheating, please report this to your Competition Manager or to the AMC HQ; these forums cannot help you.
Otherwise, please avoid public discussion of cheating. That is: no discussion of methods of cheating, no speculation about how cheating affects cutoffs, and so on --- it is not helpful to anyone, and it creates a sour atmosphere. A longer explanation is given in Seriously, please stop discussing how to cheat.
-----------------------------
15. Cutoff jokes: never allowed
Whenever the cutoffs for any major contest are released, it is very obvious when they are official. In the past, this has been achieved by the numbers being posted on the official AMC website (here) or through a post from the AMCDirector account.

You must never post fake cutoffs, even as a joke. You should also refrain from posting cutoffs that you've heard of via email, etc., because it is better to wait for the obvious official announcement. A longer explanation is given in A Treatise on Cutoff Trolling.
-----------------------------
16. Meanness: never allowed
Being mean is worse than being immature and unproductive. If another user does something which you think is inappropriate, use the Report button to bring the post to moderator attention, or if you really must reply, do so in a way that is tactful and constructive rather than inflammatory.
-----------------------------

Finally, we remind you all to sit back and enjoy the problems. :D

-----------------------------
(EDIT 2024-09-13: AoPS has asked to me to add the following item.)

Advertising paid program or service: never allowed

Per the AoPS Terms of Service (rule 5h), general advertisements are not allowed.

While we do allow advertisements of official contests (at the MAA and MATHCOUNTS level) and those run by college students with at least one successful year, any and all advertisements of a paid service or program is not allowed and will be deleted.
0 replies
v_Enhance
Jun 12, 2020
0 replies
k i Stop looking for the "right" training
v_Enhance   50
N Oct 16, 2017 by blawho12
Source: Contest advice
EDIT 2019-02-01: https://blog.evanchen.cc/2019/01/31/math-contest-platitudes-v3/ is the updated version of this.

EDIT 2021-06-09: see also https://web.evanchen.cc/faq-contest.html.

Original 2013 post
50 replies
v_Enhance
Feb 15, 2013
blawho12
Oct 16, 2017
Almost Squarefree Integers
oVlad   4
N 30 minutes ago by HeshTarg
Source: Romania Junior TST 2025 Day 1 P1
A positive integer $n\geqslant 3$ is almost squarefree if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.
4 replies
oVlad
Apr 12, 2025
HeshTarg
30 minutes ago
square root problem
kjhgyuio   1
N an hour ago by kjhgyuio
........
1 reply
kjhgyuio
an hour ago
kjhgyuio
an hour ago
A nice and easy gem off of StackExchange
NamelyOrange   2
N an hour ago by Royal_mhyasd
Source: https://math.stackexchange.com/questions/3818796/
Define $S$ as the set of all numbers of the form $2^i5^j$ for some nonnegative $i$ and $j$. Find (with proof) all pairs $(m,n)$ such that $m,n\in S$ and $m-n=1$.


Rephrased: Solve $2^a5^b-2^c5^d=1$ over $(\mathbb{N}_0)^4$, and prove that your solution(s) is/are the only one(s).
2 replies
NamelyOrange
Yesterday at 8:13 PM
Royal_mhyasd
an hour ago
Comics and triangles in perspective
srirampanchapakesan   1
N an hour ago by ohiorizzler1434
Source: Own
Let a conic intersect the sides BC, CA, AB of triangle ABC at A1,A2,B1,B2,C1,C2.

T1 is the triangle formed by A1B2, B1C2, and C1A2.

T2 is the triangle formed by A2B1, B2C1 and C2A1.

Prove that the triangles ABC, T1 and T2 are pair-wise in perspective.

Also prove that all three centers of perspective coincide.
1 reply
srirampanchapakesan
2 hours ago
ohiorizzler1434
an hour ago
9 Did I make the right choice?
Martin2001   28
N 2 hours ago by giratina3
If you were in 8th grade, would you rather go to MOP or mc nats? I chose to study the former more and got in so was wondering if that was valid given that I'll never make mc nats.
28 replies
Martin2001
Apr 29, 2025
giratina3
2 hours ago
pink mop through blue
vincentwant   8
N 2 hours ago by NoSignOfTheta
does there exist a corresponding pink mop cutoff for blue? it exists for red and i think green as well but idk about blue

if it exists what was the cutoff thsi year
8 replies
vincentwant
Yesterday at 3:48 AM
NoSignOfTheta
2 hours ago
Math Kangaroo 2025 Thread
FuturePanda   10
N 4 hours ago by mathkiddus
Are we allowed to discuss scores and problems yet? If so, we can start here.
10 replies
FuturePanda
Yesterday at 3:34 AM
mathkiddus
4 hours ago
June contests?
abbominable_sn0wman   7
N 4 hours ago by abbominable_sn0wman
are there any good/fun math contests in june? obviously arml, but anything else?
7 replies
abbominable_sn0wman
Yesterday at 1:46 AM
abbominable_sn0wman
4 hours ago
Problem 2
evt917   53
N 5 hours ago by JH_K2IMO
Source: 2024 AMC 12B #2 / AMC 10B #2
What is $10! - 7! \cdot 6!$?

$
\textbf{(A) }-120 \qquad
\textbf{(B) }0 \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }600 \qquad
\textbf{(E) }720 \qquad
$
53 replies
evt917
Nov 13, 2024
JH_K2IMO
5 hours ago
USAMO Medals
YauYauFilter   2
N 5 hours ago by Pengu14
YauYauFilter
Apr 24, 2025
Pengu14
5 hours ago
Question about AMC 10
MathNerdRabbit103   13
N Yesterday at 11:02 PM by Pengu14
Hi,

Can anybody predict a good score that I can get on the AMC 10 this November by only being good at counting and probability, number theory, and algebra? I know some geometry because I took it in school though, but it isn’t competition math so it probably doesn’t count.

Thanks.
13 replies
MathNerdRabbit103
Yesterday at 2:53 AM
Pengu14
Yesterday at 11:02 PM
2025 Math and AI 4 Girls Competition: Win Up To $1,000!!!
audio-on   65
N Yesterday at 10:45 PM by WhitePhoenix
Join the 2025 Math and AI 4 Girls Competition for a chance to win up to $1,000!

Hey Everyone, I'm pleased to announce the dates for the 2025 MA4G Competition are set!
Applications will open on March 22nd, 2025, and they will close on April 26th, 2025 (@ 11:59pm PST).

Applicants will have one month to fill out an application with prizes for the top 50 contestants & cash prizes for the top 20 contestants (including $1,000 for the winner!). More details below!

Eligibility:
The competition is free to enter, and open to middle school female students living in the US (5th-8th grade).
Award recipients are selected based on their aptitude, activities and aspirations in STEM.

Event dates:
Applications will open on March 22nd, 2025, and they will close on April 26th, 2025 (by 11:59pm PST)
Winners will be announced on June 28, 2025 during an online award ceremony.

Application requirements:
Complete a 12 question problem set on math and computer science/AI related topics
Write 2 short essays

Prizes:
1st place: $1,000 Cash prize
2nd place: $500 Cash prize
3rd place: $300 Cash prize
4th-10th: $100 Cash prize each
11th-20th: $50 Cash prize each
Top 50 contestants: Over $50 worth of gadgets and stationary


Many thanks to our current and past sponsors and partners: Hudson River Trading, MATHCOUNTS, Hewlett Packard Enterprise, Automation Anywhere, JP Morgan Chase, D.E. Shaw, and AI4ALL.

Math and AI 4 Girls is a nonprofit organization aiming to encourage young girls to develop an interest in math and AI by taking part in STEM competitions and activities at an early age. The organization will be hosting an inaugural Math and AI 4 Girls competition to identify talent and encourage long-term planning of academic and career goals in STEM.

Contact:
mathandAI4girls@yahoo.com

For more information on the competition:
https://www.mathandai4girls.org/math-and-ai-4-girls-competition

More information on how to register will be posted on the website. If you have any questions, please ask here!


65 replies
audio-on
Jan 26, 2025
WhitePhoenix
Yesterday at 10:45 PM
9 Mathpath vs. AMSP
FuturePanda   34
N Yesterday at 9:53 PM by ZMB038
Hi everyone,

For an AIME score of 7-11, would you recommend MathPath or AMSP Level 2/3?

Thanks in advance!
Also people who have gone to them, please tell me more about the programs!
34 replies
FuturePanda
Jan 30, 2025
ZMB038
Yesterday at 9:53 PM
Mathcounts state
happymoose666   34
N Yesterday at 8:51 PM by ZMB038
Hi everyone,
I just have a question. I live in PA and I sadly didn't make it to nationals this year. Is PA a competitive state? I'm new into mathcounts and not sure
34 replies
happymoose666
Mar 24, 2025
ZMB038
Yesterday at 8:51 PM
Find relation in triangle
Rushil   19
N Aug 6, 2023 by Krishijivi
Source: INMO 1992 Problem 1
In a triangle $ABC$, $\angle A = 2 \cdot \angle B$. Prove that $a^2 = b (b+c)$.
19 replies
Rushil
Oct 3, 2005
Krishijivi
Aug 6, 2023
Find relation in triangle
G H J
Source: INMO 1992 Problem 1
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Rushil
1592 posts
#1 • 2 Y
Y by Adventure10, Mango247
In a triangle $ABC$, $\angle A = 2 \cdot \angle B$. Prove that $a^2 = b (b+c)$.
This post has been edited 1 time. Last edited by Rushil, Oct 4, 2005, 5:39 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
shobber
3498 posts
#2 • 2 Y
Y by Adventure10, Mango247
Rushil wrote:
INMO 1992 Problem 1

In a triangle $ABC$ , $\angle A = 2 \times \angle B$. Prove that $a^2 = b (b+c)$
Try to search for Double Angle Theorem.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Andreas
578 posts
#3 • 3 Y
Y by AmitayasB, Adventure10, Mango247
$\frac{a}{\sin 2x} = \frac{b}{\sin x} = \frac{c}{\sin 3x}$ $\Longrightarrow$ $c = b(4\cos^2 x - 1)$ and $a = 2b\cos x$.
$b(b + c) = 4b^2\cos^2 x = a^2$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Ezbakhe Yassin
146 posts
#4 • 1 Y
Y by Adventure10
Andreas wrote:
$\frac{a}{\sin 2x} = \frac{b}{\sin x} = \frac{c}{\sin 3x}$

How did you get this one?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathmanman
1444 posts
#5 • 1 Y
Y by Adventure10
That's just the law of sines ;)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Elemennop
1421 posts
#6 • 1 Y
Y by Adventure10
$\sin{C}=\sin{(180-A-B)}=\sin{(180-3x)}=\sin{(3x)}$, Because $\sin{x}=\sin{(180-x)}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The QuattoMaster 6000
1184 posts
#7 • 1 Y
Y by Adventure10
Rushil wrote:
In a triangle $ ABC$, $ \angle A = 2 \cdot \angle B$. Prove that $ a^2 = b (b + c)$.
Sorry for awakening an old topic, but here's a neat solution that no-one here mentioned already:
Solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
earth
99 posts
#8 • 3 Y
Y by Samujjal101, Adventure10, Mango247
Hi,here is a good one ! please try out! :)
Click to reveal hidden text
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Mateescu Constantin
1842 posts
#9 • 2 Y
Y by Adventure10, Mango247
Another solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Dranzer
154 posts
#10 • 5 Y
Y by SHREYAS333, Wizard_32, PME2018, Adventure10, and 1 other user
A pure geometric solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
OmkarDivekar
2 posts
#11 • 3 Y
Y by sayanjoddar, Adventure10, Mango247
Sorry for reviving a very old thread !
We have to prove a^2=b(b+c)
Rearranging the terms we get a/b=(b+c)/a
L.H.S:-
By sine rule, a/b=sinA/sinB=2cosB ...(A=2B)
R.H.S:-
By sine rule, (b+c)/a=(sinB+sinC)/sinA=(2sin2B*cosB)/sin2B=2cosB ...(A=2B)
Hence, L.H.S=R.H.S proved.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
PhysicsMonster_01
1445 posts
#12 • 4 Y
Y by pikazag, MathematicalGiant, Adventure10, Mango247
A bit different solution with the use of trigonometry
This post has been edited 2 times. Last edited by PhysicsMonster_01, Sep 13, 2018, 7:48 AM
Reason: LaTeX
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
LoveMaths26102003
84 posts
#13 • 2 Y
Y by Adventure10, Mango247
Similarity
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ftheftics
651 posts
#14 • 2 Y
Y by Adventure10, Mango247
Suppose,$\angle B =b'$ ,$\implies \angle A =2b'$,
and,$\angle C =180-3b'$.

See that,$\frac{C}{\sin 3b'} =\frac{b}{\sin b'}$.

$\implies c=b(3-4\sin ^2 b')$.

$\implies c=b(2\cos2b'+1)$.

$\implies c^2 = bc+2bc \cos 2b'$.

Using cosine rule we have ,

$a^2=b^2+c^2 -2bc \cos 2b'$.

$\implies a^2=b^2 +bc$.

As desired.$\boxed{a^2 = b (b+c)}$
This post has been edited 1 time. Last edited by ftheftics, Feb 15, 2020, 12:38 PM
Reason: BNNn
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41957 posts
#15 • 1 Y
Y by Adventure10
Rushil wrote:
In a triangle $ABC$, $\angle A = 2 \angle B$. Prove that $a^2 = b (b+c)$.
$$\frac{a}{\sin 2B} = \frac{b}{\sin B}\Leftrightarrow  \frac{a}{ 2b}=cosB=\frac{c^2+a^2-b^2}{ 2ca}
\Leftrightarrow  a^2 = b (b+c)$$
In a triangle $ABC$, $a^2 = b (b+c)$. Prove that $\angle A = 2\angle B.$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
BestChoice123
1119 posts
#16 • 1 Y
Y by Adventure10
mathmanman wrote:
That's just the law of sines ;)

oops that link failed :(
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41957 posts
#17 • 1 Y
Y by Adventure10
BestChoice123 wrote:
mathmanman wrote:
That's just the law of sines ;)

oops that link failed :(
Law_of_Sines
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
S1281
209 posts
#18
Y by
We know that,
$a/sinA=b/sinB=c/sinC$
$a/sin2x=b /sinx=c/sin(180^{\circ{}}-3x)$
$a/2cosx=b$
$c/sin3x=b/sinx$
$c/(3sinx-4sin^3x)=b/sinx$
$c/(3-4sin^2x)=b$
$b(b+c)=b^2+bc=\frac{a^2}{4cos^2x}+\frac{a}{2cosx}(3-4sin^2x)\frac{a}{2cosx}=\frac{a^2}{4 cos^2x}4(1-sin^2x)=\frac{a^2}{4 cos^2x}4cos^2x=a^2$.
This post has been edited 2 times. Last edited by S1281, Apr 19, 2021, 12:13 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
SatisfiedMagma
458 posts
#19
Y by
I was just about the bash thing with trigonometry. But here is a geometric solution I guess.

Solution: Denote $D$ to be the foot of angle bisector of $\angle BAC$ on side $BC$. Now it is easy to notice that $\angle CAD = \angle ABD$. So we can deduce that $AC$ is tangent to $\odot(ADB)$. Applying Power of a Point along with Angle Bisector Theorem we get
\begin{align*}
b^2= \frac{ab}{b+c} \cdot a \\
\implies a^2= b(b+c)
\end{align*}which proves the desired. $\blacksquare$
This post has been edited 2 times. Last edited by SatisfiedMagma, May 14, 2022, 6:57 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Krishijivi
99 posts
#20
Y by
Let angle B=x
Then , angle A=2x
By sine rule,
a/ b=sin 2x/ sin x
cos x =a/2b
cos x=c²+a²-b²/2ac
bc²+ba²-b²-a²c=0
(c-b){b(b+c)-a²}=0
If c=b
Then also a²=b(b+c)
If b(b+c)-a²=0
a²=b(b+c)
@ Krishijivi
Z K Y
N Quick Reply
G
H
=
a