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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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hard..........
Noname23   3
N 3 minutes ago by pco
problem
3 replies
Noname23
Today at 5:42 AM
pco
3 minutes ago
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Mathhhhh
mathbetter   11
N 24 minutes ago by GreekIdiot
Three turtles are crawling along a straight road heading in the same
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?
11 replies
mathbetter
Mar 20, 2025
GreekIdiot
24 minutes ago
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Oi! These lines concur
Rg230403   19
N 38 minutes ago by cursed_tangent1434
Source: LMAO 2021 P5, LMAOSL G3(simplified)
Let $I, O$ and $\Gamma$ respectively be the incentre, circumcentre and circumcircle of triangle $ABC$. Points $A_1, A_2$ are chosen on $\Gamma$, such that $AA_1 = AI = AA_2$, and point $A'$ is the foot of the altitude from $I$ to $A_1A_2$. If $B', C'$ are similarly defined, prove that lines $AA', BB'$ and $CC'$ concurr on $OI$.
Original Version from SL
Proposed by Mahavir Gandhi
19 replies
Rg230403
May 10, 2021
cursed_tangent1434
38 minutes ago
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Arrange positive divisors of n in rectangular table!
cjquines0   43
N an hour ago by lelouchvigeo
Source: 2016 IMO Shortlist C2
Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:
[list]
[*]each cell contains a distinct divisor;
[*]the sums of all rows are equal; and
[*]the sums of all columns are equal.
[/list]
43 replies
cjquines0
Jul 19, 2017
lelouchvigeo
an hour ago
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Mathcounts state iowa
iwillregretthisnamelater   11
N Today at 4:39 AM by iwillregretthisnamelater
Ok I’m a 6th grader in Iowa who got 38 in chapter which was first, so what are the chances of me getting in nats? I should feel confident but I don’t. I have a week until states and I’m getting really anxious. What should I do? And also does the cdr count in Iowa? Because I heard that some states do cdr for fun or something and that it doesn’t count to final standings.
11 replies
iwillregretthisnamelater
Mar 20, 2025
iwillregretthisnamelater
Today at 4:39 AM
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Good Mocks for STate
Existing_Human1   1
N Today at 4:26 AM by giratina3
Hello Community!

I am wondering what are the best mocks for state, with solutions
1 reply
Existing_Human1
Yesterday at 11:52 PM
giratina3
Today at 4:26 AM
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Mathcounts STRATEGIES
Existing_Human1   27
N Today at 4:23 AM by giratina3
Hello commuinty!

I am wondering what your strategies are for mathcounts. Please note I do not mean tips. These can be for all rounds, but please specify. BTW, this is for state, but it can apply to any competition.

Ex:
Team - sit in a specific order
Target - do the easiest first
Sprint - go as fast as possible

I just made up the examples, and you will probably have better strategies, so if you want to help out, please do
27 replies
Existing_Human1
Thursday at 7:27 PM
giratina3
Today at 4:23 AM
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Problem of the week
evt917   36
N Today at 4:20 AM by giratina3
Whenever possible, I will be posting problems twice a week! They will be roughly of AMC 8 difficulty. Have fun solving! Also, these problems are all written by myself!

First problem:

$20^{16}$ has how many digits?
36 replies
evt917
Mar 5, 2025
giratina3
Today at 4:20 AM
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Tam giác nội tiếp
chunchun.math.2010   1
N Today at 2:19 AM by aidan0626
Bài 1:Cho tam giác abc nội tiếp đường tròn (o), đường cao ad (d ∈ bc). qua a kẻ đường song song với bc cắt (o) tại t. chứng minh rằng dt đi qua trọng tâm của tam giác abc.
Bài 2: Cho tứ giác ngoại tiếp abcd. p là một điểm bất kì trên cd. j, k, l lần lượt là tâm đường tròn nội tiếp của các tam giác apb, apd, bpc. chứng minh rằng ∠ajk + ∠bjl = 180°.
1 reply
chunchun.math.2010
Today at 2:06 AM
aidan0626
Today at 2:19 AM
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The daily problem!
Leeoz   2
N Yesterday at 10:06 PM by c_double_sharp
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

The first problem is:
[quote=March 21st Problem]Alice flips a fair coin until she gets 2 heads in a row, or a tail and then a head. What is the probability that she stopped after 2 heads in a row? Express your answer as a common fraction.[/quote]
2 replies
Leeoz
Yesterday at 10:01 PM
c_double_sharp
Yesterday at 10:06 PM
J
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Really Nasty MathCounts Problem
ilikemath247365   17
N Yesterday at 9:58 PM by BS2012
2019 MathCounts National Sprint #29

How many of the first $100,000$ positive integers have no single-digit prime factors?


Side note: Just HOW are they supposed to solve this in like 5 minutes?
17 replies
ilikemath247365
Mar 14, 2025
BS2012
Yesterday at 9:58 PM
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Basic Maths
JetFire008   7
N Yesterday at 9:55 PM by huajun78
Find $x$: $\sqrt{9}x=18$
7 replies
JetFire008
Yesterday at 1:19 PM
huajun78
Yesterday at 9:55 PM
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The Real Deal: Looking for Writers!
supercheetah11   6
N Yesterday at 9:45 PM by anticodon
Hello AoPS!

My name is James, and I am the editor of a math newsletter by and for kids titled "The Real Deal: A Complex Space for Kids to Discuss Math". I am looking for a few more writers willing to write an article about their favorite math problem for the coming, 6th edition of the newsletter (articles should be about 600-800 words). We have a growing readership (around 3K), and you can know that your writing will be shared with kids all over the world who also love math. If you're interested, please write me at therealdealmath@gmail.com. You can read previous issues of the newsletter at http://www.realdealmath.org.

Thank you!
6 replies
supercheetah11
Yesterday at 6:33 PM
anticodon
Yesterday at 9:45 PM
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AMC 8 Help
krish6_9   32
N Yesterday at 9:32 PM by stjwyl
Hey guys
im in new jersey a third grader who got 12 on amc 8. I want to make mop in high school and mathcounts nationals in 6th grade is that realistic how should I get better
32 replies
krish6_9
Mar 17, 2025
stjwyl
Yesterday at 9:32 PM
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Find the limit
Mathlover08092002   10
N Mar 20, 2025 by quasar_lord
Source: MTRP 2019 Class 11-Multiple Choice Question: Problem 6 :-
Find the limit $\lim \limits_{n \to \infty} \sin{n!}$
[list=1]
[*] 1
[*] 0
[*] $\frac{\pi}{4}$
[*] None of the above
[/list]
10 replies
Mathlover08092002
Apr 9, 2020
quasar_lord
Mar 20, 2025
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Find the limit
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Reveal topic tags
limits
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G H BBookmark kLocked kLocked NReply
Source: MTRP 2019 Class 11-Multiple Choice Question: Problem 6 :-
The post below has been deleted. Click to close.
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Mathlover08092002
408 posts
Mathlover08092002
#1 h  Apr 9, 2020, 6:37 PM
Y by
Find the limit $\lim \limits_{n \to \infty} \sin{n!}$
  1. 1
  2. 0
  3. $\frac{\pi}{4}$
  4. None of the above
This post has been edited 1 time. Last edited by Mathlover08092002, Apr 10, 2020, 7:53 PM
Z K Y
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starchan
1601 posts
starchan
#2 h  Apr 10, 2020, 2:45 AM
Y by
4. Is the answer
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ironsteve43
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ironsteve43
#3 h  Apr 10, 2020, 3:37 AM • 2 Y
Y by Mango247, Mango247
It is not possible to determine
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gghx
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gghx
#4 h  Apr 10, 2020, 4:06 AM • 1 Y
Y by gabrielghx
Eventually $n$ will be a multiple of 360, so the answer is 0, or option 2
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Ricky3141
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Ricky3141
#5 h  Apr 10, 2020, 6:05 AM • 1 Y
Y by SatisfiedMagma
Is sine in degrees or radians?
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Mathlover08092002
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Mathlover08092002
#6 h  Apr 14, 2020, 4:50 PM
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Show the solution
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Inconsistent
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Inconsistent
#7 h  Apr 14, 2020, 5:29 PM
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What are you talking about? This clearly diverges and alternates between 1 and -1... The answer here is D) None of the above, the limit doesn't exist. Rigorously speaking, lim sup = 1, lim inf = -1, so the limit doesn't exist. This doesn't even require calculus. Also, the sin function uses radians by default. Otherwise, the derivative of sin(x) wouldn't be cos(x) with any other scale.
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biomathematics
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biomathematics
#8 h  Apr 14, 2020, 7:13 PM
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Inconsistent wrote:
What are you talking about? This clearly diverges and alternates between 1 and -1... The answer here is D) None of the above, the limit doesn't exist. Rigorously speaking, lim sup = 1, lim inf = -1, so the limit doesn't exist. This doesn't even require calculus. Also, the sin function uses radians by default. Otherwise, the derivative of sin(x) wouldn't be cos(x) with any other scale.

It's not obvious why any of those claims you made are true.
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Inconsistent
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Inconsistent
#9 h  Apr 14, 2020, 7:43 PM
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biomathematics wrote:
Inconsistent wrote:
What are you talking about? This clearly diverges and alternates between 1 and -1... The answer here is D) None of the above, the limit doesn't exist. Rigorously speaking, lim sup = 1, lim inf = -1, so the limit doesn't exist. This doesn't even require calculus. Also, the sin function uses radians by default. Otherwise, the derivative of sin(x) wouldn't be cos(x) with any other scale.

It's not obvious why any of those claims you made are true.

Nevermind, really sorry, I thought n was continuous and ! referred to the continuous factorial (gamma function).
This post has been edited 1 time. Last edited by Inconsistent, Apr 14, 2020, 7:43 PM
Reason: edit
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lifeismathematics
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lifeismathematics
#10 h  May 15, 2022, 5:23 PM • 1 Y
Y by Mango247
answer is D
This post has been edited 2 times. Last edited by lifeismathematics, Mar 29, 2023, 3:14 PM
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quasar_lord
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quasar_lord
#12 h  Mar 20, 2025, 9:32 AM
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@above how is it clear? can you explain? i can see that the answer should be 4. but i can't prove it (normally to prove that limits do not exist we find lim sup and lim inf and show that they are not equal, but here we can't do that)
This post has been edited 1 time. Last edited by quasar_lord, Mar 20, 2025, 9:32 AM
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