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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
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k i A Letter to MSM
Arr0w   23
N Sep 19, 2022 by scannose
Greetings.

I have seen many posts talking about commonly asked questions, such as finding the value of $0^0$, $\frac{1}{0}$,$\frac{0}{0}$, $\frac{\infty}{\infty}$, why $0.999...=1$ 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 $0^0$. It is usually regarded that $0^0=1$, 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 $\frac{\infty}{\infty}$? The issue here is that $\infty$ isn't even rigorously defined in this expression. What exactly do we mean by $\infty$? Unless the example in question is put in context in a formal manner, then we say that $\frac{\infty}{\infty}$ is meaningless.

[*]What about $\frac{1}{0}$? Suppose that $x=\frac{1}{0}$. Then we would have $x\cdot 0=0=1$, absurd. A more rigorous treatment of the idea is that $\lim_{x\to0}\frac{1}{x}$ does not exist in the first place, although you will see why in a calculus course. So the point is that $\frac{1}{0}$ is undefined.

[*]What about if $0.99999...=1$? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
\begin{align*} \sum_{n=1}^{\infty} \frac{9}{10^n}&=9\sum_{n=1}^{\infty}\frac{1}{10^n}=9\sum_{n=1}^{\infty}\biggr(\frac{1}{10}\biggr)^n=9\biggr(\frac{\frac{1}{10}}{1-\frac{1}{10}}\biggr)=9\biggr(\frac{\frac{1}{10}}{\frac{9}{10}}\biggr)=9\biggr(\frac{1}{9}\biggr)=\boxed{1} \end{align*}
[*]What about $\frac{0}{0}$? 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 $0/0$ 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 $0/0$ 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 $0$, $1$ or $\infty$, 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
\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]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 $f:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}$ given by the mapping $x\mapsto \sqrt{x}$ is undefined for $x<0$. On the other hand, $1/0$ is undefined because dividing by $0$ 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 $0/0$ 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
\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]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 $-\infty\le a\le \infty$ for each $a\in\overline{\mathbb{R}}$ which means that via this order topology each subset has an infimum and supremum and $\overline{\mathbb{R}}$ is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In $\overline{\mathbb{R}}$ it is perfectly OK to say that,
\begin{align*} a + \infty = \infty + a & = \infty, & a & \neq -\infty \\ a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\ \frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\ \frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\ \frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0). \end{align*}So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,
\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]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 $\infty \times 0=0$ as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by $0$ 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
\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, $\tilde{\infty}$ means complex infinity, since we are in the complex plane now. Here's the catch: division by $0$ is allowed here! In fact, we have
\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]where $\tilde{\infty}/\tilde{\infty}$ and $0/0$ are left undefined. We also have
\begin{align*} z+\tilde{\infty}=\tilde{\infty}, \forall z\ne -\infty\\ z\times \tilde{\infty}=\tilde{\infty}, \forall z\ne 0 \end{align*}Furthermore, we actually have some nice properties with multiplication that we didn't have before. In $\mathbb{C}^*$ it holds that
\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]but $\tilde{\infty}-\tilde{\infty}$ and $0\times \tilde{\infty}$ 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 $\mathbb{C}^*$, by defining them as
\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]They behave in a similar way to the Riemann Sphere, with division by $0$ also being allowed with the same indeterminate forms (in addition to some other ones).
23 replies
Arr0w
Feb 11, 2022
scannose
Sep 19, 2022
J
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k i Marathon Threads
LauraZed   0
Jul 2, 2019
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.
0 replies
LauraZed
Jul 2, 2019
0 replies
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k i Basic Forum Rules and Info (Read before posting)
jellymoop   368
N May 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]

Here are links to more detailed versions of the rules. These are from the older forums, so you can overlook "Classroom math/Competition math only" instructions.
Posting Guidelines
Update on Basic Forum Rules
What belongs on this forum?
How do I write a thorough solution?
How do I get a problem on the contest page?
How do I study for mathcounts?
Mathcounts FAQ and resources
Mathcounts and how to learn

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.

You can also view marathons via the Marathon tag.

Think this list is incomplete or needs changes? Let the mods know and we'll take a look.
368 replies
jellymoop
May 8, 2015
harry1234
May 16, 2018
J
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Good Mocks for STate
Existing_Human1   4
N an hour ago by huajun78
Hello Community!

I am wondering what are the best mocks for state, with solutions
4 replies
Existing_Human1
Friday at 11:52 PM
huajun78
an hour ago
J
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Basic Maths
JetFire008   11
N 2 hours ago by Rice_Farmer
Find $x$: $\sqrt{9}x=18$
11 replies
JetFire008
Friday at 1:19 PM
Rice_Farmer
2 hours ago
J
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2013 Stats Sprint #28
Rice_Farmer   13
N 2 hours ago by Rice_Farmer
Is there a better way than just partitioning casework bash this?
13 replies
Rice_Farmer
Mar 17, 2025
Rice_Farmer
2 hours ago
J
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Chances at nats? Mathcounts
iwillregretthisnamelater   6
N 2 hours ago by ScoutViolet
Iowa, 38 on chapter, first in written and countdown and I’m just another person asking for nats chances.
6 replies
iwillregretthisnamelater
3 hours ago
ScoutViolet
2 hours ago
J
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Menu
pascal_1623   10
N 3 hours ago by jkim0656
On a restaurant, there are three appetizers, and four main courses. How many different dinners can be ordered if each dinner consists of one appetizer and one main course?
10 replies
pascal_1623
Aug 22, 2005
jkim0656
3 hours ago
J
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MATHCOUNTS Chapter Score Thread
apex304   109
N 3 hours ago by Mathematicalprodigy37
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ \end{tabular}$
109 replies
apex304
Mar 1, 2025
Mathematicalprodigy37
3 hours ago
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Competition Day Mindset
weihou0   27
N 4 hours ago by iwillregretthisnamelater
Hi! I'm taking the MATHCOUNTS State test on March 15. I know that there's probably not enough time left for significant skill improvement, so I wanted to ask about things I can do on competition day to help me be at my best. Most importantly, I'm curious about what mindset I should have during the test. Should I be confident and cocky, or more humble and negative? I'm just curious about what preparations prior to the test help the most. Thanks!
27 replies
weihou0
Mar 9, 2025
iwillregretthisnamelater
4 hours ago
J
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Confusion about 2 counting strategies
Spacepandamath13   2
N 4 hours ago by mathelvin
What is the difference between PIE (Principle of Inclusion and Exclusion) and Complementary Counting?
2 replies
Spacepandamath13
4 hours ago
mathelvin
4 hours ago
J
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1/a+1/b+1/c=6/7 - some fun
236factorial   10
N 4 hours ago by Jaxman8
If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{6}{7}$, where a, b, and c are positive integers, what is the smallest value of a+b+c?
10 replies
236factorial
Feb 11, 2006
Jaxman8
4 hours ago
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Click to reveal hidden content
236factorial   13
N 5 hours ago by KF329
If there are a total of 951 hide tags on the basic forum, and 231 do not say "click to reveal hidden content". How many hide tags without the labelling "click to reveal hidden content" must be posted consecutively for the percentage of these hide tags to rise to 25%?

Please hide your answers :D
13 replies
236factorial
Aug 22, 2005
KF329
5 hours ago
J
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spotted in blogroll
Major_Monogram   3
N Yesterday at 8:17 PM by Demetri
I saw this on the AoPS Blogroll. closing it, the page worked normally.
3 replies
Major_Monogram
Yesterday at 7:37 PM
Demetri
Yesterday at 8:17 PM
J
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Python exit() module decriptions appear as "undefined"
SoaringHigh   8
N Yesterday at 7:59 PM by Major_Monogram
Summary of the problem: When using exit() (or quit()) in the Python windows on AoPS the "Description" and "To fix" options show up as "undefined"
sample program
Page URL: N/A
Steps to reproduce:
1. Use the AoPS Python module to execute the exit() or quit() functions in a program. (try running the sample program)
Expected behavior: The "Description" and "To fix" sections give a description of SystemExit
Frequency: Always
Operating system(s): Windows 11 Home
Browser(s), including version: Microsoft Edge 130.0.2849.46
Additional information: N/A
8 replies
SoaringHigh
Oct 22, 2024
Major_Monogram
Yesterday at 7:59 PM
J
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Help about topics on alcumus
GreenBanana666   10
N Yesterday at 7:40 PM by Major_Monogram
My ratio basics score used to be 99 a 5 days ago. But it is 84 now. What and why did that happen?


Now after one hour and zero questions done on Ratio basics, it has risen by 5 rating. :ewpu:
10 replies
GreenBanana666
Oct 29, 2024
Major_Monogram
Yesterday at 7:40 PM
J
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The avatars are not consistent
Craftybutterfly   34
N Yesterday at 4:38 PM by Demetri
Summary of the problem: The avatars are not consistent
Page URL: idk
Steps to reproduce:
1. change your avatar
2.reload a topic you posted in
3. do #2 to a different topic with your post in it
Expected behavior: Avatars are the same
Frequency: 100%
Operating system(s): MacOS
Browser(s), including version: Chrome latest version
Additional information: refreshing does not help, neither does logging out and in
34 replies
Craftybutterfly
Mar 20, 2025
Demetri
Yesterday at 4:38 PM
J
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J
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k Number of posts is not consistent
LawofCosine   3
N Mar 21, 2025 by LawofCosine
The number of posts is not consistent. see attachments. Thank you.
3 replies
LawofCosine
Mar 21, 2025
LawofCosine
Mar 21, 2025
J
Number of posts is not consistent
G H J
Reveal topic tags
number of posts
L
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
LawofCosine
624 posts
LawofCosine
#1 h  Mar 21, 2025, 1:16 AM
Y by
The number of posts is not consistent. see attachments. Thank you.
Attachments:
Z Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
stjwyl
1232 posts
stjwyl
#2 h  Mar 21, 2025, 1:22 AM
Y by
It's because you posted
At the time of the previous post you had 604 posts
then you posted again upping your post count by one

refreshing should fix that
this isnt a major bug or anything
Z Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Demetri
1106 posts
Demetri
#3 h  Mar 21, 2025, 1:24 AM
Y by
LawofCosine wrote:
The number of posts is not consistent. see attachments. Thank you.
Refreshing should fix this :)
Quote:
If a refresh makes it go away 100% of the time, then it isn't a bug, but rather a glitch.
This is classified as a glitch, not a bug. In fact its probably not a glitch at all since the way aops works is that it doesn't constantly update posts if you have already loaded them. So posts that were saved before will be outdated until you refresh.
Z Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
LawofCosine
624 posts
LawofCosine
#4 h  Mar 21, 2025, 1:27 AM • 3 Y
Y by Demetri, aidan0626, skronkmonster
Thanks! Refreshing does help.
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a