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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
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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 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
Division involving difference of squares
BR1F1SZ   2
N 2 minutes ago by NumberzAndStuff
Source: Austria National MO Part 1 Problem 4
Determine all integers $n$ that can be written in the form
\[
n = \frac{a^2 - b^2}{b},
\]where $a$ and $b$ are positive integers.

(Walther Janous)
2 replies
BR1F1SZ
May 5, 2025
NumberzAndStuff
2 minutes ago
Common tangent to diameter circles
Stuttgarden   4
N 6 minutes ago by zuat.e
Source: Spain MO 2025 P2
The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.
4 replies
Stuttgarden
Mar 31, 2025
zuat.e
6 minutes ago
IMO Shortlist 2014 N5
hajimbrak   61
N 7 minutes ago by Markas
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.

Proposed by Belgium
61 replies
1 viewing
hajimbrak
Jul 11, 2015
Markas
7 minutes ago
Number theory
falantrng   39
N 8 minutes ago by Markas
Source: RMM 2018 D2 P4
Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.
39 replies
falantrng
Feb 25, 2018
Markas
8 minutes ago
Past USAMO Medals
sdpandit   2
N Today at 6:27 AM by sdpandit
Does anyone know where to find lists of USAMO medalists from past years? I can find the 2025 list on their website, but they don't seem to keep lists from previous years and I can't find it anywhere else. Thanks!
2 replies
sdpandit
May 8, 2025
sdpandit
Today at 6:27 AM
Geo is back??
GoodMorning   137
N Today at 5:58 AM by Siddharthmaybe
Source: 2023 USAJMO Problem 2/USAMO Problem 1
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.

Proposed by Holden Mui
137 replies
GoodMorning
Mar 23, 2023
Siddharthmaybe
Today at 5:58 AM
TMC (Tompkins Math Contest) Online for Middle Schoolers
loona_stan   0
Today at 3:12 AM
Source: https://othsmao.github.io/TMC/index.html
Hi AOPS Community!

The Tompkins High School Mu Alpha Theta Chapter would like to present to you the TMC(Tompkins Math Contest).

Here is some info about us and the contest:

Who: We are a highschool math club based in Katy, Texas who are looking to create positive impact in the math contest community. This contest is made for middle schoolers (grades 6-8).
When: 5/24. More info about specific schedule will be released closer to the contest date.
Where: It will be hosted online!
Prizes: We were kindly given prizes to distribute by AOPS and there will possibly be other prizes.
Content: The contest will feature a mix of TMSCA, AMC 8, AMC 10, and AIME problems, totaling 50 questions. The contest features tiebreakers with AIME level questions. This means anyone from any skill level should feel free to participate!

You can sign up using this link:https://docs.google.com/forms/d/e/1FAIpQLSc-tMw6fff_FMKwRVWtc91M54Us7rBtLK6rSnM7MMeCV5iqqA/viewform

And see more about us on our website: https://othsmao.github.io/TMC/index.html

We hope to see lots of yall on contest day!!
0 replies
loona_stan
Today at 3:12 AM
0 replies
[CASH PRIZES] IndyINTEGIRLS Spring Math Competition
Indy_Integirls   0
Today at 2:36 AM
[center]IMAGE

Greetings, AoPS! IndyINTEGIRLS will be hosting a virtual math competition on May 25,
2024 from 12 PM to 3 PM EST.
Join other woman-identifying and/or non-binary "STEMinists" in solving problems, socializing, playing games, winning prizes, and more! If you are interested in competing, please register here![/center]

----------

[center]Important Information[/center]

Eligibility: This competition is open to all woman-identifying and non-binary students in middle and high school. Non-Indiana residents and international students are welcome as well!

Format: There will be a middle school and high school division. In each separate division, there will be an individual round and a team round, where students are grouped into teams of 3-4 and collaboratively solve a set of difficult problems. There will also be a buzzer/countdown/Kahoot-style round, where students from both divisions are grouped together to compete in a MATHCOUNTS-style countdown round! There will be prizes for the top competitors in each division.

Problem Difficulty: Our amazing team of problem writers is working hard to ensure that there will be problems for problem-solvers of all levels! The middle school problems will range from MATHCOUNTS school round to AMC 10 level, while the high school problems will be for more advanced problem-solvers. The team round problems will cover various difficulty levels and are meant to be more difficult, while the countdown/buzzer/Kahoot round questions will be similar to MATHCOUNTS state to MATHCOUNTS Nationals countdown round in difficulty.

Platform: This contest will be held virtually through Zoom. All competitors are required to have their cameras turned on at all times unless they have a reason for otherwise. Proctors and volunteers will be monitoring students at all times to prevent cheating and to create a fair environment for all students.

Prizes: At this moment, prizes are TBD, and more information will be provided and attached to this post as the competition date approaches. Rest assured, IndyINTEGIRLS has historically given out very generous cash prizes, and we intend on maintaining this generosity into our Spring Competition.

Contact & Connect With Us: Follow us on Instagram @indy.integirls, join our Discord, follow us on TikTok @indy.integirls, and email us at indy@integirls.org.

---------
[center]Help Us Out

Please help us in sharing the news of this competition! Our amazing team of officers has worked very hard to provide this educational opportunity to as many students as possible, and we would appreciate it if you could help us spread the word!
0 replies
Indy_Integirls
Today at 2:36 AM
0 replies
MATHCOUNTS halp
AndrewZhong2012   22
N Today at 1:49 AM by Math-lover1
I know this post has been made before, but I personally can't find it. I qualified for mathcounts through wildcard in PA, and I can't figure out how to do those last handful of states sprint problems that seem to be one trick ponies(2024 P28 and P29 are examples) They seem very prevalent recently. Does anyone have advice on how to figure out problems like these in the moment?
22 replies
AndrewZhong2012
Mar 5, 2025
Math-lover1
Today at 1:49 AM
camp/class recommendations for incoming freshman
walterboro   5
N Yesterday at 11:58 PM by franklin2013
hi guys, i'm about to be an incoming freshman, does anyone have recommendations for classes to take next year and camps this summer? i am sure that i can aime qual but not jmo qual yet. ty
5 replies
walterboro
Yesterday at 6:45 PM
franklin2013
Yesterday at 11:58 PM
Summer internships/research opportunists in STEM
o99999   6
N Yesterday at 10:50 PM by Pengu14
Hi, I am a current high school student and was looking for internships and research opportunities in STEM. Do you guys know any summer programs that do research such as RSI, but for high school freshmen that are open?
Thanks.
6 replies
o99999
Apr 22, 2020
Pengu14
Yesterday at 10:50 PM
HCSSiM results
SurvivingInEnglish   65
N Yesterday at 9:51 PM by NoSignOfTheta
Anyone already got results for HCSSiM? Are there any point in sending additional work if I applied on March 19?
65 replies
SurvivingInEnglish
Apr 5, 2024
NoSignOfTheta
Yesterday at 9:51 PM
9 JMO<200?
DreamineYT   1
N Yesterday at 5:44 PM by Shan3t
Just wanted to ask
1 reply
DreamineYT
Yesterday at 5:37 PM
Shan3t
Yesterday at 5:44 PM
9 ARML Location
deduck   42
N Yesterday at 5:40 PM by llddmmtt1
UNR -> Nevada
St Anselm -> New Hampshire
PSU -> Pennsylvania
WCU -> North Carolina


Put your USERNAME in the list ONLY IF YOU WANT TO!!!! !!!!!

I'm going to UNR if anyone wants to meetup!!! :D

Current List:
Iowa
UNR
PSU
St Anselm
WCU
42 replies
deduck
May 6, 2025
llddmmtt1
Yesterday at 5:40 PM
IMOC 2020 G5 concyclic wanted, parallelogram and concurrent related
parmenides51   5
N Apr 17, 2023 by Mahdi_Mashayekhi
Source: https://artofproblemsolving.com/community/c6h2254883p17398793
Let $O, H$ be the circumcentor and the orthocenter of a scalene triangle $ABC$. Let $P$ be the reflection of $A$ w.r.t. $OH$, and $Q$ is a point on $\odot (ABC)$ such that $AQ, OH, BC$ are concurrent. Let $A'$ be a points such that $ABA'C$ is a parallelogram. Show that $A', H, P, Q$ are concylic.

(ltf0501).
5 replies
parmenides51
Sep 1, 2020
Mahdi_Mashayekhi
Apr 17, 2023
IMOC 2020 G5 concyclic wanted, parallelogram and concurrent related
G H J
G H BBookmark kLocked kLocked NReply
Source: https://artofproblemsolving.com/community/c6h2254883p17398793
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parmenides51
30652 posts
#1
Y by
Let $O, H$ be the circumcentor and the orthocenter of a scalene triangle $ABC$. Let $P$ be the reflection of $A$ w.r.t. $OH$, and $Q$ is a point on $\odot (ABC)$ such that $AQ, OH, BC$ are concurrent. Let $A'$ be a points such that $ABA'C$ is a parallelogram. Show that $A', H, P, Q$ are concylic.

(ltf0501).
This post has been edited 1 time. Last edited by parmenides51, Dec 15, 2022, 7:33 PM
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amar_04
1915 posts
#2 • 1 Y
Y by Gaussian_cyber
Perform a $-\sqrt{HA\cdot HD}$ Inversion around $H$. We get the following Equivalent Problem.

$\textbf{INVERTED PROBLEM:-}$ $ABC$ be a triangle with Circumcenter $O$ and Incenter $I$. Let $\overline{OI}\cap\odot(ABC)=T$ and $\odot(AIT)\cap\odot(ABC)=K$. Let $A'$ be the reflection of $A$ over $\overline{IO}$. and $D$ be the point where the Incircle of $\Delta ABC$ touches $\overline{BC}$. Then $A',D,K$ are collinear.

By Radical Axis Theorem on $\odot(ABC),\odot(BIC),\odot(AIT)$ we get that $\overline{AK},\overline{IO},\overline{BC}$ are concurrent. Now the rest follows from #4 here. $\blacksquare$
This post has been edited 1 time. Last edited by amar_04, Sep 1, 2020, 8:48 PM
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WolfusA
1900 posts
#3 • 1 Y
Y by bin_sherlo
Let $a,b,c$ be complex numbers lying on unit circle such that they are vertices of the triangle. Then
$$h=a+b+c, a'=b+c-a,\ p=\frac{h\overline{a}}{\overline{h}}=bc\cdot\frac{a+b+c}{ab+bc+ca}.$$From $R$ definition
$$\left(\overline{r}=\frac{b+c-r}{bc}\ \wedge\ \frac{\overline{r}}{\overline{h}}=\frac{r}{h}\right)\iff r=\frac{a(b+c)(a+b+c)}{a^2+2a(b+c)+bc}.$$$Q$ definition implies $$\overline{r}=\frac{a+q-r}{aq}\iff q=\frac{a-r}{a\cdot\overline{r}-1}=\frac{(a+b)(a+c)-(b+c)^2}{a^2(b+c)^2-(a+b)(a+c)bc}\cdot abc.$$Moving on...
$$h-p=\frac{a(b+c)(a+b+c)}{ab+bc+ca},\ h-a'=2a,$$$$q-a'=\frac{(a-b)(a-c)(b+c)(ab+bc+ca)}{a^2(b+c)^2-(a+b)(a+c)bc},$$$$q-p=\frac{bc(b+c)(a^2-bc)(a^2+2a(b+c)+bc)}{(ab+bc+ca)(a^2(b+c)^2-(a+b)(a+c)bc)}.$$Therefore $$\frac{h-p}{h-a'}\cdot\frac{q-a'}{q-p}=\frac{(a+b+c)(a-b)(a-c)(b+c)(ab+bc+ca)}{2bc(a^2-bc)(a^2+2a(b+c)+bc)}=\overline{\left(\frac{h-p}{h-a'}\cdot\frac{q-a'}{q-p}\right)}\square.$$#1726
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ltf0501
191 posts
#4 • 1 Y
Y by amar_04
amar_04 wrote:
Perform a $-\sqrt{HA\cdot HD}$ Inversion around $H$. We get the following Equivalent Problem.

$\textbf{INVERTED PROBLEM:-}$ $ABC$ be a triangle with Circumcenter $O$ and Incenter $I$. Let $\overline{OI}\cap\odot(ABC)=T$ and $\odot(AIT)\cap\odot(ABC)=K$. Let $A'$ be the reflection of $A$ over $\overline{IO}$. and $D$ be the point where the Incircle of $\Delta ABC$ touches $\overline{BC}$. Then $A',D,K$ are collinear.

By Radical Axis Theorem on $\odot(ABC),\odot(BIC),\odot(AIT)$ we get that $\overline{AK},\overline{IO},\overline{BC}$ are concurrent. Now the rest follows from #4 here. $\blacksquare$

That's how this problem was produced. :)
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USJL
540 posts
#5 • 2 Y
Y by gnoka, SerdarBozdag
I didn't know how to do the original TST problem, but I somehow manage to solve this one :o
I think my solution is kind of cute so I'm posting it here.

Note that $B,H,C,A'$ are concyclic. Therefore it suffices to show that $BC,A'H,PQ$ are concurrent (by, say, power of a point theorem). Let $A_1$ be the antipedal point of $A$ w.r.t. $\odot(ABC)$. Since the intersections $AA_1\cap BC$ and $A'H\cap BC$ are symmetric w.r.t. the midpoint $M$ of $BC$, by the butterfly theorem (or more precisely, a slightly more generalized statement of it), it suffices to show that $AQ\cap BC$ and $PA_1\cap BC$ are symmetric w.r.t. $M$. Note that $M$ is also a midpoint of $HA_1$ and both $OH, PA_1$ are perpendicular to $AP$. Therefore the lines $OH$ and $PA_1$ are symmetric w.r.t. $M$, which gives the desired result.
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Mahdi_Mashayekhi
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#6
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Let $OH$ meet $BC$ at $T$. Let $AH$ and $AO$ meet $ABC$ at $S$ and $A''$. Let $M$ be midpoint of $BC$.
Note that $BHCA'$ is cyclic so by Radical Axis Theorem we need to prove $HA'$ and $PQ$ meet at $BC$. Let $PQ$ and $A'H$ meet $BC$ at $X$ and $Y$. Note that $\frac{BX}{XC} = \frac{BQ}{QC}.\frac{BP}{PC}$ and $\frac{BY}{YC} = \frac{BH}{HC}.\frac{BA'}{A'C}$ so we need to prove $\frac{BQ}{QC}.\frac{BP}{PC} = \frac{BS}{SC}.\frac{AC}{AB}$ or $\frac{BQ}{QC}.\frac{AB}{AC} = \frac{BS}{SC}.\frac{PC}{PB}$ or $\frac{TC}{TB} = \frac{CA''}{BA''}.\frac{PC}{PB}$. Let $T'$ be reflection of $T$ across $M$. we need to prove $\frac{T'C}{T'B} = \frac{CA''}{BA''}.\frac{PC}{PB}$ or in fact we need to prove $T',A'',P$ are collinear. Note that $M$ is midpoint of $HA''$ so $THT'A''$ is parallelogram so $TO || T'A''$ and $PA'' \perp AP \perp TO$ so $PA'' || TO$ so $P,A'',T'$ are collinear as wanted.
we're Done.
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