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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 1, 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
J
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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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I need your help
Mr_adjective   22
N 2 hours ago by Dragon_Yang
how do you use aops?
22 replies
Mr_adjective
Thursday at 6:37 PM
Dragon_Yang
2 hours ago
J
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an elegant geometric configuration
Mr_adjective   1
N 2 hours ago by OronSH
Let \(ABC\) be a triangle with incenter \(I\), and let \(AD\) be an altitude (where \(D\) lies on \(BC\)). Denote by \(A'\) the point diametrically opposite to \(A\) on the circumcircle of \(\triangle ABC\). Prove that \(\angle ADI + \angle AIA' = 180^\circ\).
1 reply
Mr_adjective
2 hours ago
OronSH
2 hours ago
J
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Bogus Proof Marathon
pifinity   7760
N 3 hours ago by IamCurlyLizard39
Hi!
I'd like to introduce the Bogus Proof Marathon.

In this marathon, simply post a bogus proof that is middle-school level and the next person will find the error. You don't have to post the real solution :P

Use classic Marathon format: 
[hide=P#]a1b2c3[/hide]
[hide=S#]a1b2c3[/hide]


Example posts:

P(x)
-----
S(x)
P(x+1)
-----
Let's go!! Just don't make it too hard!
7760 replies
pifinity
Mar 12, 2018
IamCurlyLizard39
3 hours ago
J
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Should I reset Alcumus?
SunnieBunnie   17
N 4 hours ago by SunnieBunnie
My Alcumus rating is pretty low, 59.5, and I only did about 90% of my problems correct out of 328. Should I reset?
17 replies
SunnieBunnie
Yesterday at 6:52 PM
SunnieBunnie
4 hours ago
J
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Private Forum for Geometry Study
fossasor   240
N 4 hours ago by SunnieBunnie
Hi all,

I've recently been trying to improve my skill at competition math, but I consistently struggle with geometry. Since doing math is more fun and motivating with others, I've created the GeoPrepClub. This is a private forum to increase geometry skill and help others increase theirs: we'll have problems, marathons, and much more. This is similar to forums such as "AMC8 Prep Buddies" by PatTheKing, but is focused exclusively on this subject. We welcome all skill levels and hope those with greater mathematical knowledge can assist those lacking in it.

If this sounds interesting to you, sing up below and I'll let you know once I've added you. Although the forum may not have much now, that's because I've only just released it, and I hope once I build a community, It will be a very useful and motivating space for those interested in improving their geometry. The link is
here.

I look forward to seeing you all in the forum!
240 replies
fossasor
Jun 10, 2025
SunnieBunnie
4 hours ago
J
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Proving Algebra
Justbrick   0
4 hours ago
Prove that for all positive real numbers $x, y, z > 0$ such that $x + y + z = 3$, we have:
\[ \frac{(1 - x)^2}{1 - x^4} + \frac{(1 - y)^2}{1 - y^4} + \frac{(1 - z)^2}{1 - z^4} \ge 0. \]
0 replies
Justbrick
4 hours ago
0 replies
J
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The 24 Game No Postfarming Edition.
maxamc   36
N 5 hours ago by PikaPika999
Use the numbers $1,2,3,4,5,6,7,8,9,10$ (you must use all numbers) and the operations $+,-,\cdot,\div$ ONLY to create all positive integers in order. I will repeat 1 more time: ONLY these 4 operations (no concatenation or anything like totient etc.)

This is a real challenge compared to all postfarming attempts over the years.

Solutions
36 replies
maxamc
Yesterday at 4:10 PM
PikaPika999
5 hours ago
J
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Math Counts
sudarya   6
N 6 hours ago by DetectiveBanana25
Hello guys!



My account name is sudarya and I am working hard to go to Math Counts Nationals with my state (Wisconsin). I am having some trouble lately at some state problems, and I kind of need some encouragement from you guys. This is my only hope of making it to Nationals, and I really trust you guys to make this dream a reality for me. Thank you so much!


6 replies
sudarya
Yesterday at 3:42 PM
DetectiveBanana25
6 hours ago
J
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9 Pythagorean Triples
ZMB038   85
N Today at 12:33 AM by PikaPika999
Please put some of the ones you know, and try not to troll/start flame wars! Thank you :D
85 replies
ZMB038
May 19, 2025
PikaPika999
Today at 12:33 AM
J
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Scores on Contests for Ya'll to submit! (everyone be as honest as possible)
NEILDASEAL_12345   112
N Today at 12:25 AM by DreamineYT
What are yall's most recent scores on mathCON, AMC 8 and other things? I just wanna know how I'm holding up and if my scores are good! This is the end of my first year in contest math, and I've always been gifted, but can someone tell me if these scores are good? MATHCON: 240
AMC 8: 19 QUESTIONS
AMC 10: 16 QUESTIONS
btw I'm going to 7th grade

EDIT: Y'all I mean 16 questions right on the AMC 10, I'm way too lazy to scroll and find the number of points lol
EDIT: Math Kangaroo: I forgot the score, but I got like 30th National
112 replies
NEILDASEAL_12345
Jul 10, 2025
DreamineYT
Today at 12:25 AM
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Amc8 mock
Provyfx   21
N Yesterday at 9:44 PM by JerryZYang
1. find the area of the circle that circumscribes a right triangle whose legs are lengths of $6cm$ and $10cm$.
2. find the sum of all the $4$ - digit positive numbers with no zero digit
3. Let $ABCD$ be a trapezoid with $BC||AD$ and $AB = BC = CD =0.5AD$. Determine $<ACD$
4. When Jack wakes up everyday, he picks a sock. There is $1$ blue sock, $2$ white socks, $3$ red socks and $4$ yellow socks. Because Jack is biased in his picking. The probability of him picking out a blue sock is $4$ times as likely as a yellow socks, $3$ times as likely as a red sock, and $2$ times as likely as a white sock. What is the probability of him picking a yellow or white sock?

I used formula $A$ $=$ $abc/4R$ in problem 1
to get circumradius as $5$
21 replies
Provyfx
Yesterday at 6:14 PM
JerryZYang
Yesterday at 9:44 PM
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Random but useful theorems
booking   102
N Yesterday at 9:19 PM by Hanruz
There have been all these random but useful theorems
Please post any theorems you know, random or not, but please say whether they are random or not.
I'll start give an example:
Random
I am just looking for some theorems to study.
102 replies
booking
Jul 16, 2025
Hanruz
Yesterday at 9:19 PM
J
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no answer problem
Mathichian   18
N Yesterday at 9:07 PM by codeninja
if you chose a random answer for this problem, what is the probability that you are correct?
A)25% B)50% C)60% D)25%
18 replies
Mathichian
Jul 22, 2025
codeninja
Yesterday at 9:07 PM
J
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9 How many perfect squares do you know?
mymelody1234   9
N Yesterday at 8:37 PM by sadas123
I've memorized 30 perfect squares :)
I meant to say consecutive squares oops
9 replies
mymelody1234
Yesterday at 7:33 PM
sadas123
Yesterday at 8:37 PM
J
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J
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An algebra math problem
AVY2024   6
N Apr 18, 2025 by Roger.Moore
Solve for a,b
ax-2b=5bx-3a
6 replies
AVY2024
Apr 8, 2025
Roger.Moore
Apr 18, 2025
J
An algebra math problem
G H J
Reveal topic tags
algebra
nmtc
equation
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G H BBookmark kLocked kLocked NReply
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AVY2024
30 posts
AVY2024
#1 h  Apr 8, 2025, 10:55 AM
Y by
Solve for a,b
ax-2b=5bx-3a
Z K Y
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Inaaya
512 posts
Inaaya
#2 h  Apr 8, 2025, 1:40 PM • 1 Y
Y by Exponent11
There are 3 variables and one equation, this thing can theoretically have infinite solution sets
Z K Y
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sp0rtman00000
2 posts
sp0rtman00000
#3 h  Apr 11, 2025, 11:03 AM
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On the R or on the N?
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Pengu14
652 posts
Pengu14
#5 h  Apr 11, 2025, 3:24 PM
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Inaaya wrote:
There are 3 variables and one equation, this thing can theoretically have infinite solution sets

I believe they’re looking for a and b such that the equation holds for ALL x.
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Pengu14
652 posts
Pengu14
#6 h  Apr 11, 2025, 3:25 PM
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AVY2024 wrote:
Solve for a,b
ax-2b=5bx-3a

We have a=5b and 3a=2b. The only solution is a=b=0.
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Runner1600
12 posts
Runner1600
#7 h  Apr 11, 2025, 3:56 PM
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Pengu14 wrote:
AVY2024 wrote:
Solve for a,b
ax-2b=5bx-3a

We have a=5b and 3a=2b. The only solution is a=b=0.

That is what I got
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Roger.Moore
5 posts
Roger.Moore
#8 h  Apr 18, 2025, 6:03 PM
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The x there are parameter?
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