ka April Highlights and 2025 AoPS Online Class Information
jlacosta0
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)!
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
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
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
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
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
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
I have seen many posts talking about commonly asked questions, such as finding the value of ,,,, why or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
[list]
[*]Firstly, the case of . It is usually regarded that , not because this works numerically but because it is convenient to define it this way. You will see the convenience of defining other undefined things later on in this post.
[*]What about ? The issue here is that isn't even rigorously defined in this expression. What exactly do we mean by ? Unless the example in question is put in context in a formal manner, then we say that is meaningless.
[*]What about ? Suppose that . Then we would have , absurd. A more rigorous treatment of the idea is that does not exist in the first place, although you will see why in a calculus course. So the point is that is undefined.
[*]What about if ? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
[*]What about ? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[/list]
Hopefully all of these issues and their corollaries are finally put to rest. Cheers.
2nd EDIT (6/14/22): Since I originally posted this, it has since blown up so I will try to add additional information per the request of users in the thread below.
INDETERMINATE VS UNDEFINED
What makes something indeterminate? As you can see above, there are many things that are indeterminate. While definitions might vary slightly, it is the consensus that the following definition holds: A mathematical expression is be said to be indeterminate if it is not definitively or precisely determined. So how does this make, say, something like indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that is an indeterminate form.
But we need to more specific, this is still ambiguous. An indeterminate form is a mathematical expression involving at most two of , or , obtained by applying the algebraic limit theorem (a theorem in analysis, look this up for details) in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being calculated. This is why it is called indeterminate. Some examples of indeterminate forms are etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function given by the mapping is undefined for . On the other hand, is undefined because dividing by is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context.
WHEN THE WATERS GET MUDDIED
So with this notion of indeterminate and undefined, things get convoluted. First of all, just because something is indeterminate does not mean it is not undefined. For example is considered both indeterminate and undefined (but in the context of a limit then it is considered in indeterminate form). Additionally, this notion of something being undefined also means that we can define it in some way. To rephrase, this means that technically, we can make something that is undefined to something that is defined as long as we define it. I'll show you what I mean.
One example of making something undefined into something defined is the extended real number line, which we define as So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let for each which means that via this order topology each subset has an infimum and supremum and is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In it is perfectly OK to say that, So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined, So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by is undefined still! Is there a place where it isn't? Kind of. To do this, we can extend the complex numbers! More formally, we can define this extension as which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, means complex infinity, since we are in the complex plane now. Here's the catch: division by is allowed here! In fact, we have where and are left undefined. We also have Furthermore, we actually have some nice properties with multiplication that we didn't have before. In it holds that but and are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with , by defining them as They behave in a similar way to the Riemann Sphere, with division by also being allowed with the same indeterminate forms (in addition to some other ones).
Due to excessive spam and inappropriate posts, we have locked the Prealgebra and Beginning Algebra threads.
We will either unlock these threads once we've cleaned them up or start new ones, but for now, do not start new marathon threads for these subjects. Any new marathon threads started while this announcement is up will be immediately deleted.
ki Basic Forum Rules and Info (Read before posting)
jellymoop368
NMay 16, 2018
by harry1234
f(Reminder: Do not post Alcumus or class homework questions on this forum. Instructions below.)f
Welcome to the Middle School Math Forum! Please take a moment to familiarize yourself with the rules.
Overview:
[list]
[*] When you're posting a new topic with a math problem, give the topic a detailed title that includes the subject of the problem (not just "easy problem" or "nice problem")
[*] Stay on topic and be courteous.
[*] Hide solutions!
[*] If you see an inappropriate post in this forum, simply report the post and a moderator will deal with it. Don't make your own post telling people they're not following the rules - that usually just makes the issue worse.
[*] When you post a question that you need help solving, post what you've attempted so far and not just the question. We are here to learn from each other, not to do your homework. :P
[*] Avoid making posts just to thank someone - you can use the upvote function instead
[*] Don't make a new reply just to repeat yourself or comment on the quality of others' posts; instead, post when you have a new insight or question. You can also edit your post if it's the most recent and you want to add more information.
[*] Avoid bumping old posts.
[*] Use GameBot to post alcumus questions.
[*] If you need general MATHCOUNTS/math competition advice, check out the threads below.
[*] Don't post other users' real names.
[*] Advertisements are not allowed. You can advertise your forum on your profile with a link, on your blog, and on user-created forums that permit forum advertisements.
[/list]
As always, if you have any questions, you can PM me or any of the other Middle School Moderators. Once again, if you see spam, it would help a lot if you filed a report instead of responding :)
Marathons!
Relays might be a better way to describe it, but these threads definitely go the distance! One person starts off by posting a problem, and the next person comes up with a solution and a new problem for another user to solve. Here's some of the frequently active marathons running in this forum:
[list][*]Algebra
[*]Prealgebra
[*]Proofs
[*]Factoring
[*]Geometry
[*]Counting & Probability
[*]Number Theory[/list]
Some of these haven't received attention in a while, but these are the main ones for their respective subjects. Rather than starting a new marathon, please give the existing ones a shot first.
Source: Own , Mock Thailand Mathematic Olympiad P1
Let be a scalene triangle with point and on the plane such that is an equilateral . Let intersect and at and respectively and intersect and at and respectively .
Prove that
In equilateral triangle , let . Point lies on segment such that . Let and be the circumcenter and incenter of triangle , respectively. Let and be the circumcenter and incenter of triangle , respectively. Suppose that lines and intersect at point . Find the length of .
Berkeley mini Math Tournament is a math competition hosted for middle school students once a year. Students compete in multiple rounds: individual round, team round, puzzle round, and relay round.
BmMT 2025 Online will be held on June 7th, and registration is OPEN! Registration is $8 per student. Our website https://berkeley.mt/events/bmmt-2025-online/ has more details about the event, past tests to practice with, and frequently asked questions. We look forward to building community and inspiring students as they explore the world of math!
3 out of 4 of the rounds are completed with a team, so it’s a great opportunity for students to work together. Beyond getting more comfortable with math and becoming better problem solvers, our team is preparing some fun post-competition activities!
Registration is open to students in grades 8 or below. You do not have to be local to the Bay Area or California to register for BmMT Online. Students may register as a team of 1, but it is beneficial to compete on a team of at least 3 due to our scoring guideline and for the experience.
We hope you consider attending, or if you are a parent or teacher, that you encourage your students to think about attending BmMT. Thank you, and once again find more details/register at our website,https://berkeley.mt.
Summer STEM Success Series with Fikky Dosunmu!
Empower Your Learning. Sharpen Your Skills. Crush Your Goals.
Who Am I?
Hi! I’m Fikky Dosunmu, an incoming freshman at Carnegie Mellon University’s School of Computer Science, and was admitted to several other top institutions, including the California Institute of Technology.
This Summer’s Course Offerings (8–10 Weeks)
SAT Math Mastery
From Algebra to Advanced Word Problems — Learn test-taking strategies, avoid common traps, and aim for a perfect math score. USACO Bronze Bootcamp
Jump into competitive programming! Master problem types in C++/Java, develop algorithmic thinking, and prep for Silver-level promotion. Single Variable Calculus
Tackle limits, derivatives, integrals, and real-world applications with guided instruction tailored to AP/college-level rigor.
Why Learn with Me? Selected Finalist – CMU’s prestigious Summer Academy for Math and Science (SAMS)
Run a YouTube channel with 70K+ views and 700+ subscribers, where I teach math and coding to students around the world with clarity and passion 1st Place – HPE CodeWars Advanced 2025
Solved over 500 competitive programming problems on platforms like Codeforces and USACO
Years of Science Bowl and Science Olympiad Experience AP Scholar with 5s in AP Calculus BC, Computer Science A, Statistics, Physics, Chemistry, Psychology
Scored 800 on the SAT Math section (Perfect Score) Hands-on coding and teaching experience through projects, contests, and club leadership
Format
Each course runs for 8 to 10 weeks, tailored to student pacing.
Classes held online via Zoom or Google Meet — flexible scheduling.
Price ranges from 160 to 200 (USD) per course (based on duration & customization).
First 2 classes are FREE — no commitment required!
Money-back guarantee if you’re not satisfied after the second class.
If you are interested in class, shoot me an email (specify which class) at nucleusdosunmu96@gmail.com
Hi! So I was playing Connect4 with my friends the other day and I wondered: how many "legal" arrangements of Connect4 can be reached at the ending position?
We assume that we do not stop the game when there is a four in a row, and we have 21 red pieces and 21 yellow pieces. We also drop the pieces one by one into a standard 7 by 6 board. We can start the game with any color piece.
https://en.wikipedia.org/wiki/Connect_Four
Initial Thoughts
This problem seems easy at first; the number of arrangments is simply However, I quickly saw that some boards
OOOOOOO
OOOOOOO
OOOOOOO
OOOOOOO
OOOOOOO
OOOOOOO
were impossible to construct by just dropping pieces one by one like a normal game.
Attempt to use one-to-one correspondences
After I realized that my Initial Thoughts weren't going to work, I tried to use one-to-one correspondences. I represented the columns as ABCDEFG from left to right and represented dropping the red/yellow pieces as a string of length 21 of these letters. This seemed to solve my problem, but new roadblocks popped up.
Roadblock 1 There is more than one way to represent a certain configuration using this correspondence. A quick example
red pieces fill all the left 3 columns, yellow pieces fill all the right 3 columns
shows that we overcount some configurations by using this method.
Roadblock 2 Even if we didn't overcount, we still need to account for the fact that the total number of A, B, C... over both of the strings have to each equal 7. The amount of cases (1 A goes to the red pieces, 6 As go to the yellow pieces,...) would be very difficult to calculate, even using a computer.
Alright, I was thinking about why 0.999...=1 one day, and remembered something from learning calculus. Technically, subtracting 0.999... from 1 gives 0.000...0001 (infinitely many zeros). This is really close to zero, however I will denote as 0+, as it indeed is greater than 0, even by the smallest margin. Now take the differences (1-0.999...999) and add them up infinitely many times. Should it be zero? No because of the 0+, its a bit greater than 0, so adding it up infinitely many times would be greater than 0... Whats's wrong with my reasoning?
I hit my 1234th post! (I think I missed it, I'm kinda late, :oops_sign:)
But here's a puzzle for you all! Try to create the numbers 1 through 25 using the numbers 1, 2, 3, and 4! You are only allowed to use addition, subtraction, multiplication, division, and parenthesis. If you're post #1, try to make 1. If you're post #2, try to make 2. If you're post #3, try to make 3, and so on. If you're a post after 25, then I guess you can try to make numbers greater than 25 but you can use factorials, square roots, and that stuff. Have fun!