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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
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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
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What belongs on this forum?
How do I write a thorough solution?
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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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Another geometry problem
Rice_Farmer   0
16 minutes ago
In triangle $ABC,AB=6,AC=8,$ and $BC=10.M$ and $N$ are the midpoints of $AB$ and $BC$ respectively. Ray $CM$ is extended past $M$ to a point $Y$ such that the circumcircle of $AMY$ is tangent to $AN.$Find the area of triangle $NAY.$
0 replies
1 viewing
Rice_Farmer
16 minutes ago
0 replies
J
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What is the length of AP?
tqm1062010   1
N an hour ago by tqm1062010
Let ABC be an equilateral triangle with a side length of 3. Points M and N are on sides BC and CA, respectively, such that BM = CN = 2. Point P is on side AB such that NP is perpendicular to AM. What is the length of AP ?
1 reply
tqm1062010
2 hours ago
tqm1062010
an hour ago
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Number Finding!
Kushagra2012   19
N an hour ago by Kushagra2012
Find My Numbers!

The challenge for you is to identify three distinct whole numbers where the result of adding them together is the same as the result of multiplying them together.
19 replies
Kushagra2012
Yesterday at 1:35 AM
Kushagra2012
an hour ago
J
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9 MathDash
booking   48
N an hour ago by Andyluo
If you pay for MathDash, specifically lessons, could you please give feedback below?
48 replies
booking
Jul 6, 2025
Andyluo
an hour ago
J
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Khan Academy vs. MSM vs. Alcumus
a.zvezda   13
N an hour ago by Spacepandamath13
Which one is best for AMC8 prep? I want to improve geo and algebra and some C&P.
13 replies
a.zvezda
Yesterday at 6:17 PM
Spacepandamath13
an hour ago
J
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6-7 number pairs
hellohi321   9
N 2 hours ago by Spacepandamath13
Define a 6-7 number pair to be a pair of consecutive positive integers such that when read out loud one after another, you hear the phrase "six seven". For example, 76, 77 is a 6-7 number pair since it is pronounced "seventy-six seventy-seven". Find a closed form for the number of 6-7 pairs in which both terms are less than $10^n$

Sorry if this has a lot of brainrot lol. I thought of this while listening to brainrotted kids as a summer volunteer
9 replies
hellohi321
Jul 27, 2025
Spacepandamath13
2 hours ago
J
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The 24 Game Only Basic Operations
maxamc   72
N 2 hours ago by K1mchi_
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.)

No using ChatGPT, programming, Wolfram|Alpha etc. to solve a number.

Solutions
72 replies
maxamc
Jul 25, 2025
K1mchi_
2 hours ago
J
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Inequalities
sqing   2
N 2 hours ago by smileapple
Let $ a,b \geq 0, 2a +b^2=1 . $ Prove that
$$ \frac{\sqrt{a+b}}{a(b+1)} \geq \sqrt{2} $$Let $ a,b \geq 0, a^2 +2b^2 =2 . $ Prove that
$$ \frac{\sqrt{a+2b}}{a(b+1)} \geq \frac{9}{8\sqrt{2}} $$Let $ a,b \geq 0, 2a^2 +b^2 =1 . $ Prove that
$$\dfrac{a^2}{a+b}\leq\frac{1}{\sqrt{2}} $$$$ \frac{7}{10}>\frac{a+b}{a^2+b^2+1} \geq\frac{\sqrt{2}}{3} $$
2 replies
sqing
Yesterday at 12:30 PM
smileapple
2 hours ago
J
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Nice Inequality
MathRook7817   4
N 2 hours ago by sqing
Nice question I found:
4 replies
MathRook7817
Yesterday at 5:08 PM
sqing
2 hours ago
J
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Smart Mathathon
Kushagra2012   4
N 3 hours ago by JerryZYang
The Smart Mathathon!

This is my smart mathathon! Try to create problems on this forum topic.
Example:

You could post this: Do $4576 + 3457 * 0 - 0$ in the smartest way possible (don't post this problem).

The post below the other post MUST include 2 answers from a above problem. First and second people who post do not have problems to solve, but they should make one.

And begin!
4 replies
Kushagra2012
4 hours ago
JerryZYang
3 hours ago
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MATHCOUNTS Geometry Problem
Owinner   3
N 3 hours ago by Leeoz
This problem is the MATHCOUNTS 2019 Chapter Sprint #29. I have solved it before using mass points and area techniques, but that was after many failures. I actually tried it recently, and I wasn't able to do it. I would like to know if there is another solution instead of mass points. A solution using mass points is fine as well.
3 replies
Owinner
Yesterday at 3:33 AM
Leeoz
3 hours ago
J
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A tale of dragons - demo
ethanhansummerfun   1
N 3 hours ago by Ynsg
This is a part of a series that I’ve been working on. If y’all like this type of problem I can publish the whole set.

4. In the land of $\sqrt{-1}$, the continent is divided into $10$ regions, as shown. Dragon island, far in the northeast, has just birthed $12$ dragolings. Dragolings are very territorial. Any dragoling next to another one of the same element will immediately attempt to annihilate one another and lay waste to the entire continent. You must assign each dragoling to its own region. Adjacent sides are NOT ok, but touching corners are. An old master informs you just before you convene your council that an extremely rare wind dragoling may be coming. He informs you that it must have one ice, fire, AND lightning adjacent to it or else it will be bored and also destroy the whole continent. Dismissing it as a myth, your council informs you that you have enough power to create $2$ new regions by cutting any already existing region in half by connecting points not adjacent to each other such that the line drawn passes through exactly $1$ region. As you ask your scouts what they saw, you realize in horror that they said they scouted $4$ fire, $4$ ice, $3$ lightning, and $1$ mysterious white dragon, the legendary wind dragon. This is not a Zelda reference even though my pfp says otherwise.

You have 10 minutes. (not really) Which 2 regions should you bisect to save the continent?
1 reply
ethanhansummerfun
Yesterday at 4:36 AM
Ynsg
3 hours ago
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My First Math Tournament
dragnin   31
N 4 hours ago by superhuman233
I'm back with a 2nd math story! :jump: Do you remember your first math competition? Was it scary? :noo: Did you brag a lot after it even though you were probably not as smart as you say? :flex: I think we all improved a lot since then! :icecream:
Click to reveal hidden text
Epilogue:
Click to reveal hidden text

Here is my other story if anyone is interested: https://artofproblemsolving.com/community/c3t968896f3h2640331_the_best_day_of_my_life_so_far_an_inspiring_story

Thank you for reading!

31 replies
dragnin
Sep 12, 2021
superhuman233
4 hours ago
J
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Smart Math
Kushagra2012   4
N 4 hours ago by Kushagra2012
Smart Calculations

Here is a set of problems!

$1$) What is the fastest way to solve this with paper and pencil only?: $4000 + 603 + 97 + 100 + 400 * 0 + 1200 + 0 + 0 + 100 + 10000 + 0 + 500$.

$2$) Calculate $7532 * 3 + 6784 * 0 + 0$.

$3$) A car travels at $60$ miles per hour for $2$ hours and $30$ minutes. Then, the car increases its speed by $10$ miles per hour and continues for another hour and $45$ minutes. How far did the car travel in total?
4 replies
Kushagra2012
6 hours ago
Kushagra2012
4 hours ago
J
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Easy Function
Darealzolt   1
N Jun 6, 2025 by alexheinis
Let \( f(x+y) = f(x^2y)\) for all real numbers \(x,y\), hence find the value of \(f(3)\) if \(f(2023)=26\).
1 reply
Darealzolt
Jun 6, 2025
alexheinis
Jun 6, 2025
J
Easy Function
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Darealzolt
96 posts
Darealzolt
#1 h  Jun 6, 2025, 8:47 AM
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Let \( f(x+y) = f(x^2y)\) for all real numbers \(x,y\), hence find the value of \(f(3)\) if \(f(2023)=26\).
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alexheinis
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alexheinis
#2 h  Jun 6, 2025, 8:50 AM
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Take $y=0$ then $f(x)=f(0)$ and $f$ is constant. Hence $f(3)=26$.
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