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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
J
H
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
H
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
H
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
H
9 MathDash
booking   13
N 3 minutes ago by deeptisidana
If you pay for MathDash, specifically lessons, could you please give feedback below?
13 replies
booking
Jul 6, 2025
deeptisidana
3 minutes ago
J
H
9 Easiest math competition
a.zvezda   43
N 37 minutes ago by min.jonathan
MOEMS, Math Kangaroo, and Beestar are to me the easiest. :P
43 replies
a.zvezda
Jul 15, 2025
min.jonathan
37 minutes ago
J
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Is anybody else in ICPS???????
AmberFirefly33   5
N an hour ago by min.jonathan
I recently got into ICPS which is a math program for advanced math people. Is anybody else in there? BTW it's in New York!
5 replies
AmberFirefly33
Jul 18, 2025
min.jonathan
an hour ago
J
H
Bogus Proof Marathon
pifinity   7734
N an hour ago by Amargolese
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!
7734 replies
pifinity
Mar 12, 2018
Amargolese
an hour ago
J
H
2500 VIMC mathematics competition
OWOW   47
N an hour ago by HappyOreoGuineaPig24
Hello! This is my second math competition I made up, this time, set in the year 2500 on a Venus base in the upper atmosphere.
This is basically AMC8 for Venusians.

2500 VIMC

What is the sum of all perfect squares less than 2500 in the form $9p^2$ where $p$ is prime?

Equilateral triangle $ABC$ has Area of $6$. If point $D$ is the midpoint of $AB$, then line $L$ is $CD$, which cuts the equilateral triangle in half. What is the sum of the perimeters of $ACD$ and $BCD$?

A Quadnumber is a number whose digits sum to $4$. How many positive 3-digit quadnumbers are there? (ex. $310$ is a quadnumber because it is positive, 3-digit, and the digits sum to $4$)

A wheel has $100$ equal sections with the numbers from $1$ to $100$ written on them. If you spin the wheel $N$ times, what is the probability that you will land on a prime number on any of the $N$ spins? (Clarification: Probability of Landing on a prime AT LEAST ONCE in the N spins)

Car A travels at a constant $10$ mph. Car B is a probabilistic car, and at the start of every hour, it gets the option for a $10\%$ chance of traveling at $5$ mph, an $80\%$ chance of traveling at $10$ mph, and a $10\%$ chance of traveling at $15$ mph. Once Car B decides on a speed, it acts like a normal car for the rest of the hour with a constant speed at which it picked at the start of the hour, and then the same options happen at the start of the next hour, and so on. Car A and Car B leave at the same time, and they both go in the same direction. What is the probability that Car B is right next to Car A (same speed, same position) for only $3$ hours in the next 6 hours?

If $x \equiv 2\pmod{3}$, and $x \equiv 3\pmod{4}$, and $x \equiv 4\pmod{5}$, what is the least positive value of $x$.

If $f(x)=3x-1$ and $g(x)$ is $x^2-3x+3$, what is $f(g(5)-f(4))+g(f(3)-g(2))+1$?



This competition is shorter than the Mars version (MIMC 2200) which was 10 questions. But this one I believe is harder than the MIMC. (designed to be, anyway) Btw here is the entire MIMC competition in case you missed it a couple weeks ago.



$25^2-10^2=$?

If ball A costs $\$1.50$, and ball B costs $\$1.75$, what is the average cost per ball if you buy $10$ of ball A and $4$ of ball B? Please write as an improper fraction

How many divisors does $2200^3$ have?

If $f(2200-k)=10(k-2)$, then what is $f(f(2^{11}))$?

If three distinct positive primes, $(p,q,r)$, sum to 50, what is the greatest possible value of $(r^p)-(q^p)$ such that $p<q<r$?

What positive integer $x$ satisfies: $x^5-(x+4)^3+10^{(5-3)}=0$?

If a circle with radius $1$ is inscribed in a square, which is inscribed in another circle, what is the area of the outer circle minus the area of the inner circle?

What is the sum of $b$ and $c$ such that the roots of $x^2+bx+c=0$ sum to $11c$ and multiply to $12b-7$?

If the operation $a\#b = \dfrac{(a-b+1)^2}{(b-a+1)^2}$, what is the largest possible value of $(3\#x)$ given that $x$ is a positive integer?

How many 4-digit numbers have the property that the sum of the thousands and tens digits combined is the square of the sum of the hundreds and ones digits combined?

Pls write all solutions like this:
ex. S1 (Contest: either VIMC or MIMC)
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Completed problems (I'm not going to show solutions here so other people can solve it)
MIMC: 1,2,3,4,5,6,7,8,9 (9/10 done)
VIMC: 1,3,6,7 (4/7 done)
47 replies
OWOW
Jul 16, 2025
HappyOreoGuineaPig24
an hour ago
J
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The 24 Game, but with a twist!
PikaPika999   395
N 2 hours ago by K1mchi_
So many people know the 24 game, where you try to create the number 24 from using other numbers, but here's a twist:

You can only use the number 24 (up to 5 times) to try to make other numbers :)

the limit is 5 times because then people could just do $\frac{24}{24}+\frac{24}{24}+\frac{24}{24}+...$ and so on to create any number!

honestly, I feel like with only addition, subtraction, multiplication, and division, you can't get pretty far with this, so you can use any mathematical operations!

Banned functions
395 replies
PikaPika999
Jul 1, 2025
K1mchi_
2 hours ago
J
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Random but useful theorems
booking   46
N 2 hours ago by Spacepandamath13
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.
46 replies
booking
Jul 16, 2025
Spacepandamath13
2 hours ago
J
H
Fun Problem
vmene   1
N 2 hours ago by sadas123
Here's a fun, self-made problem. Let $a$ be the number of subsets of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ (including the set itself and the null set), let $b$ be the number of ways you can create a triangle from a regular octagon, and let $c$ be the sum of the squares of the roots of the quadratic $x^2=5x-6$. Find $(a+b+c)^2$. (You may use a calculator for the final expression.)
Answer:
Solution:
1 reply
vmene
3 hours ago
sadas123
2 hours ago
J
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Divisibility
Studying_geometry   1
N 2 hours ago by mithu542
If $ n \in \mathbb{Z}^+ $ such that $ n-3 \mid n^3-3 $
Find the maximum value of $ n $
1 reply
Studying_geometry
6 hours ago
mithu542
2 hours ago
J
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4,000 Official Posts: Goodbye MSM
pingpongmerrily   18
N 5 hours ago by Oshawoot
This marks my 4,000th official post, which will probably be my last milestone of 1,000. These posts have come in a lot of ways: music threads in AtAC, spamming cutoff predictions throughout November, yapping about botting on Alcumus, or sharing solutions to MathCounts questions. Over the last year or so of my forum activity, I've learned a lot, met some new people, and had some fun.
I do not think I will hit another multiple of 1,000, primarily because next year I will be starting high school and will have significantly less time to spend here. I will miss a lot about Middle School, but I will primarily be talking about Math in this post. As I get busier and busier, I realized I may have missed my window to grind. I've already taken my last MathCounts test, completed my last AMC-8, and it won't be long before I'm done with the AMC-10 too. The competition I'll miss most is probably MathCounts, just because I'll never experience anything like it again. The thrill of going up to compete in CDR, spending time lounging in the Westin in Downtown DC, speeding through Sprint questions, only to go back and realize I had sillied over a quarter of them. (If you want to read my long, yet unfinished MathCounts yap train, click this link and go to the "2025 MathCounts Recaps" hidetag in the OP).
Anyway, enough yapping. Here are some problems I made up (I'll add more later after I finish my writing problem).

1. jlcong is flipping coins. If the number of heads ever exceeds the number of tails, he starts a flamewar. Once he flips the coin 11 times, he gets bored and stops. What is the probability that he starts a flamewar?
2. Find $1989^{69^{1434}}\pmod{42}$
3. Call a set $\text{normal}$ if it is not a member of itself. Call a set $\text{abnormal}$ if it is a member of itself. Call a set $\text{questionable}$ if it is impossible to determine whether or not it is normal. For example, the set of all abnormal sets is questionable. Prove that the set of all questionable sets is normal.
4. Find an expression for $\phi(p^2)$, where $p$ is a randomly selected prime.
18 replies
pingpongmerrily
Jul 7, 2025
Oshawoot
5 hours ago
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Website to learn math
hawa   163
N 5 hours ago by K1mchi_
Hi, I'm kinda curious what website do yall use to learn math, like i dont find any website thats fun to learn math
163 replies
hawa
Apr 9, 2025
K1mchi_
5 hours ago
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Nice Geometry Problem
aruether   2
N Today at 11:20 AM by Studying_geometry
A nice geometry problem my friends:

https://youtu.be/1NlRkuwbBGo

Have fun!
2 replies
aruether
Jul 19, 2025
Studying_geometry
Today at 11:20 AM
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Divisibility
Justbrick   22
N Today at 5:25 AM by Mr_adjective
Prove that for all integers $n$.
Prove that \( n^5 - 5n^3 + 4n \) is divisible by 120
22 replies
Justbrick
Yesterday at 1:35 AM
Mr_adjective
Today at 5:25 AM
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gcd of n(n+1)(n+2)(n+3)(n+4)
Marius_Avion_De_Vanatoare   8
N Yesterday at 11:59 PM by vincentwant
Find the greatest common divisor of the numbers: $1\cdot 2\cdot 3 \cdot 4 \cdot 5$ and $2\cdot 3 \cdot 4 \cdot 5 \cdot 6 \dots 2025 \cdot 2026 \cdot 2027 \cdot 2028 \cdot 2029$.
8 replies
Marius_Avion_De_Vanatoare
Jun 14, 2025
vincentwant
Yesterday at 11:59 PM
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easy problem
yt12   3
N Jun 8, 2025 by trangbui
In a box, there are 15 cards numbered from 1 to 15. Four cards are randomly drawn from the box at the same time. Find the probability that two of the selected cards are multiples of 2 and the other two are multiples of 3.
3 replies
yt12
Jun 6, 2025
trangbui
Jun 8, 2025
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easy problem
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yt12
769 posts
yt12
#1 h  Jun 6, 2025, 3:58 PM
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In a box, there are 15 cards numbered from 1 to 15. Four cards are randomly drawn from the box at the same time. Find the probability that two of the selected cards are multiples of 2 and the other two are multiples of 3.
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yt12
769 posts
yt12
#2 h  Jun 8, 2025, 1:45 PM
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help me please
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Soupboy0
628 posts
Soupboy0
#3 h  Jun 8, 2025, 2:40 PM
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Try using PIE for this
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trangbui
840 posts
trangbui
#4 h  Jun 8, 2025, 2:55 PM
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Helpful information  The number of ways to choose \(k\) items from a set of \(n\) items is given by the binomial coefficient \({n \choose k}=\frac{n!}{k!(n-k)!}\). The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.
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