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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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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
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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.
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What belongs on this forum?
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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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Abelkonkurransen 2025 4b
Lil_flip38   1
N 6 minutes ago by Lil_flip38
Source: abelkonkurransen
Determine the largest real number \(C\) such that
$$\frac{1}{x}+\frac{1}{2y}+\frac{1}{3z}\geqslant C$$for all real numbers \(x,y,z\neq 0\) satisfying the equation
$$\frac{x}{yz}+\frac{4y}{xz}+\frac{9z}{xy}=24$$
1 reply
Lil_flip38
22 minutes ago
Lil_flip38
6 minutes ago
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Fun issue about Euler’s function
luutrongphuc   0
18 minutes ago
Let $p$ is a prime number and $n,\alpha$ are positive integers. Prove that there exist infinitely $a$ such that $\phi(a),\phi(a+1),…,\phi(a+n)$ are divisible by $p^{\alpha}$
0 replies
luutrongphuc
18 minutes ago
0 replies
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Combinatorial NT involving sum of fractional parts
Photaesthesia   14
N 19 minutes ago by jjjppp110
Source: 2025 China Mathematical Olympiad Day 1 Problem 3
Let \(a_1, a_2, \ldots, a_n\) be integers such that \(a_1 > a_2 > \cdots > a_n > 1\). Let \(M = \operatorname{lcm} \left( a_1, a_2, \ldots, a_n \right)\). For any finite nonempty set $X$ of positive integers, define \[ f(X) = \min_{1 \leqslant i \leqslant n} \sum_{x \in X} \left\{ \frac{x}{a_i} \right\}. \]Such a set $X$ is called minimal if for every proper subset $Y$ of it, $f(Y) < f(X)$ always holds.

Suppose $X$ is minimal and $f(X) \geqslant \frac{2}{a_n}$. Prove that \[ |X| \leqslant f(X) \cdot M. \]
14 replies
1 viewing
Photaesthesia
Nov 27, 2024
jjjppp110
19 minutes ago
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Mathhhhh
mathbetter   1
N 22 minutes ago by togrulhamidli2011
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?
1 reply
mathbetter
25 minutes ago
togrulhamidli2011
22 minutes ago
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Combi counting
Caasi_Gnow   0
4 hours ago
Find the number of different ways to arrange seven people around a circular meeting table if A and B must sit together and C and D cannot sit next to each other. (Note: the order for A and B might be A,B or B,A)
0 replies
Caasi_Gnow
4 hours ago
0 replies
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Mathcounts state iowa
iwillregretthisnamelater   1
N Today at 5:23 AM by Dutchb
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.
1 reply
iwillregretthisnamelater
Today at 4:55 AM
Dutchb
Today at 5:23 AM
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I need help!-tpulak
tpulak   11
N Today at 4:58 AM by mdk2013
Hello peoples:
I need help on a problem. If you know the solution, could you please post it, and tell me how to do it? Here is the problem:
"The product (66)(9)(22)(39) has a prime factorization of the form (2A)(3B)(11C)(13D). What is the value of ac-bd? "

{Note: 2a actually represents "2 to the power of a", same goes for 3b,11c, & 13D}

PLEASE HELP ME! :?: :!: :!: :!:
11 replies
1 viewing
tpulak
Nov 30, 2007
mdk2013
Today at 4:58 AM
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a problem
Bummer12345   4
N Today at 4:09 AM by FunBrightStage
Alice and Bob play a game where Alice starts with $3$ MathJuice bottles and Bob starts with $2$ MathJuice bottles. An unfair coin is then flipped, with probability $\frac{2}{3}$ of landing heads. If the coin lands heads, Alice gives Bob a bottle; otherwise, Bob gives Alice a bottle. This process repeats until someone runs out of bottles.

(a): What is the probability that Bob will lose all of his bottles before Alice does?
(b): What is the expected number of times the coin has been flipped by the time the game ends?

Source: Own
4 replies
Bummer12345
Yesterday at 8:00 PM
FunBrightStage
Today at 4:09 AM
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AMC 8 Help
krish6_9   29
N Today at 3:56 AM by jkim0656
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
29 replies
krish6_9
Mar 17, 2025
jkim0656
Today at 3:56 AM
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MATHCOUNTS on ESPN
rrusczyk   17
N Today at 3:08 AM by DreamineYT
ESPN noon EST - the Countdown round of Nationals.

(Disclaimer: yours truly is an 'analyst' for the broadcast.)
17 replies
rrusczyk
May 27, 2003
DreamineYT
Today at 3:08 AM
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probability pentagon contains center
JohnStuckey   6
N Today at 3:06 AM by jkim0656
Here's a cute problem:

Consider a regular pentagon. Choose 3 points along the perimeter of the pentagon, and form a triangle with those 3 points. What is the probability that this triangle contains the center of the pentagon.
6 replies
JohnStuckey
Mar 17, 2025
jkim0656
Today at 3:06 AM
J
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Tangent Spheres in a plane
ReticulatedPython   8
N Today at 1:10 AM by ReticulatedPython
Three mutually tangent spheres with radius $6$ rest on a plane. A sphere with radius $10$ is tangent to all of them, but does not intersect nor lie on the plane. A sphere with radius $r$ lies on the plane, and is tangent to all three spheres with radius $6.$ Compute the shortest distance between the sphere with radius $r$ and the sphere with radius $10.$

Source: Own
8 replies
ReticulatedPython
Yesterday at 9:44 PM
ReticulatedPython
Today at 1:10 AM
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Need some help on this problem please?
tpulak   11
N Today at 12:46 AM by jkim0656
What is the number of distinct lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral?

Please tell me an explanation if you can solve
11 replies
tpulak
Feb 18, 2010
jkim0656
Today at 12:46 AM
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Angle problem
FlyingDragon21   22
N Today at 12:40 AM by Bnn81351
In Isosceles triangle ABC where AB equals AC and point D lies on line AB, line CD splits line AB so that AD equals BC. If angle BAC is 20 degrees, what is the measure of angle DCA?
22 replies
FlyingDragon21
Tuesday at 11:36 PM
Bnn81351
Today at 12:40 AM
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Euler's function
luutrongphuc   0
Yesterday at 5:18 AM
Prove that for each pair of positive integers $a,m$: there exists an integer $n >1$ such that: $v_2(n!) \equiv a (mod m)$
0 replies
luutrongphuc
Yesterday at 5:18 AM
0 replies
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Euler's function
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22 posts
luutrongphuc
#1 h  Yesterday at 5:18 AM
Y by
Prove that for each pair of positive integers $a,m$: there exists an integer $n >1$ such that: $v_2(n!) \equiv a (mod m)$
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