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

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

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 1, 2025
0 replies
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k i A Letter to MSM
Arr0w   23
N Sep 19, 2022 by scannose
Greetings.

I have seen many posts talking about commonly asked questions, such as finding the value of $0^0$, $\frac{1}{0}$,$\frac{0}{0}$, $\frac{\infty}{\infty}$, why $0.999...=1$ or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
[list]
[*]Firstly, the case of $0^0$. It is usually regarded that $0^0=1$, not because this works numerically but because it is convenient to define it this way. You will see the convenience of defining other undefined things later on in this post.

[*]What about $\frac{\infty}{\infty}$? The issue here is that $\infty$ isn't even rigorously defined in this expression. What exactly do we mean by $\infty$? Unless the example in question is put in context in a formal manner, then we say that $\frac{\infty}{\infty}$ is meaningless.

[*]What about $\frac{1}{0}$? Suppose that $x=\frac{1}{0}$. Then we would have $x\cdot 0=0=1$, absurd. A more rigorous treatment of the idea is that $\lim_{x\to0}\frac{1}{x}$ does not exist in the first place, although you will see why in a calculus course. So the point is that $\frac{1}{0}$ is undefined.

[*]What about if $0.99999...=1$? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
\begin{align*} \sum_{n=1}^{\infty} \frac{9}{10^n}&=9\sum_{n=1}^{\infty}\frac{1}{10^n}=9\sum_{n=1}^{\infty}\biggr(\frac{1}{10}\biggr)^n=9\biggr(\frac{\frac{1}{10}}{1-\frac{1}{10}}\biggr)=9\biggr(\frac{\frac{1}{10}}{\frac{9}{10}}\biggr)=9\biggr(\frac{1}{9}\biggr)=\boxed{1} \end{align*}
[*]What about $\frac{0}{0}$? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[/list]
Hopefully all of these issues and their corollaries are finally put to rest. Cheers.

2nd EDIT (6/14/22): Since I originally posted this, it has since blown up so I will try to add additional information per the request of users in the thread below.

INDETERMINATE VS UNDEFINED

What makes something indeterminate? As you can see above, there are many things that are indeterminate. While definitions might vary slightly, it is the consensus that the following definition holds: A mathematical expression is be said to be indeterminate if it is not definitively or precisely determined. So how does this make, say, something like $0/0$ indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that $0/0$ is an indeterminate form.

But we need to more specific, this is still ambiguous. An indeterminate form is a mathematical expression involving at most two of $0$, $1$ or $\infty$, obtained by applying the algebraic limit theorem (a theorem in analysis, look this up for details) in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being calculated. This is why it is called indeterminate. Some examples of indeterminate forms are
\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function $f:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}$ given by the mapping $x\mapsto \sqrt{x}$ is undefined for $x<0$. On the other hand, $1/0$ is undefined because dividing by $0$ is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context.

WHEN THE WATERS GET MUDDIED

So with this notion of indeterminate and undefined, things get convoluted. First of all, just because something is indeterminate does not mean it is not undefined. For example $0/0$ is considered both indeterminate and undefined (but in the context of a limit then it is considered in indeterminate form). Additionally, this notion of something being undefined also means that we can define it in some way. To rephrase, this means that technically, we can make something that is undefined to something that is defined as long as we define it. I'll show you what I mean.

One example of making something undefined into something defined is the extended real number line, which we define as
\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let $-\infty\le a\le \infty$ for each $a\in\overline{\mathbb{R}}$ which means that via this order topology each subset has an infimum and supremum and $\overline{\mathbb{R}}$ is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In $\overline{\mathbb{R}}$ it is perfectly OK to say that,
\begin{align*} a + \infty = \infty + a & = \infty, & a & \neq -\infty \\ a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\ \frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\ \frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\ \frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0). \end{align*}So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,
\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define $\infty \times 0=0$ as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by $0$ is undefined still! Is there a place where it isn't? Kind of. To do this, we can extend the complex numbers! More formally, we can define this extension as
\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, $\tilde{\infty}$ means complex infinity, since we are in the complex plane now. Here's the catch: division by $0$ is allowed here! In fact, we have
\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]where $\tilde{\infty}/\tilde{\infty}$ and $0/0$ are left undefined. We also have
\begin{align*} z+\tilde{\infty}=\tilde{\infty}, \forall z\ne -\infty\\ z\times \tilde{\infty}=\tilde{\infty}, \forall z\ne 0 \end{align*}Furthermore, we actually have some nice properties with multiplication that we didn't have before. In $\mathbb{C}^*$ it holds that
\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]but $\tilde{\infty}-\tilde{\infty}$ and $0\times \tilde{\infty}$ are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with $\mathbb{C}^*$, by defining them as
\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]They behave in a similar way to the Riemann Sphere, with division by $0$ also being allowed with the same indeterminate forms (in addition to some other ones).
23 replies
Arr0w
Feb 11, 2022
scannose
Sep 19, 2022
J
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k i Marathon Threads
LauraZed   0
Jul 2, 2019
Due to excessive spam and inappropriate posts, we have locked the Prealgebra and Beginning Algebra threads.

We will either unlock these threads once we've cleaned them up or start new ones, but for now, do not start new marathon threads for these subjects. Any new marathon threads started while this announcement is up will be immediately deleted.
0 replies
LauraZed
Jul 2, 2019
0 replies
J
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k i Basic Forum Rules and Info (Read before posting)
jellymoop   368
N May 16, 2018 by harry1234
f (Reminder: Do not post Alcumus or class homework questions on this forum. Instructions below.) f
Welcome to the Middle School Math Forum! Please take a moment to familiarize yourself with the rules.

Overview:
[list]
[*] When you're posting a new topic with a math problem, give the topic a detailed title that includes the subject of the problem (not just "easy problem" or "nice problem")
[*] Stay on topic and be courteous.
[*] Hide solutions!
[*] If you see an inappropriate post in this forum, simply report the post and a moderator will deal with it. Don't make your own post telling people they're not following the rules - that usually just makes the issue worse.
[*] When you post a question that you need help solving, post what you've attempted so far and not just the question. We are here to learn from each other, not to do your homework. :P
[*] Avoid making posts just to thank someone - you can use the upvote function instead
[*] Don't make a new reply just to repeat yourself or comment on the quality of others' posts; instead, post when you have a new insight or question. You can also edit your post if it's the most recent and you want to add more information.
[*] Avoid bumping old posts.
[*] Use GameBot to post alcumus questions.
[*] If you need general MATHCOUNTS/math competition advice, check out the threads below.
[*] Don't post other users' real names.
[*] Advertisements are not allowed. You can advertise your forum on your profile with a link, on your blog, and on user-created forums that permit forum advertisements.
[/list]

Here are links to more detailed versions of the rules. These are from the older forums, so you can overlook "Classroom math/Competition math only" instructions.
Posting Guidelines
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What belongs on this forum?
How do I write a thorough solution?
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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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On a generalization of a^3+b^3+c^3-3abc
Sivin   1
N 32 minutes ago by Sivin
We note that $x_1^2+x_2^2-2x_1x_2=(x_1-x_2)^2$ and
$${x_1^3+x_2^3+x_3^3-3x_1x_2x_3=(x_1+x_2+x_3)(x_1^2+x_2^2+x_3^2-x_1x_2-x_2x_3-x_3x_1)}$$are reducible polynomials in $\mathbb{C}.$ However, $x_1^4+x_2^4+x_3^4+x_4^4-4x_1x_2x_3x_4$ is an irreducible polynomial in $\mathbb{C}.$ So what are all the $n$ such that the polynomial:
$$f_n(x_1,x_2,\dots,x_n)=x_1^n+x_2^n+\dots+x_n^n-nx_1x_2\dots x_n$$is reducible in $\mathbb{C}$?
1 reply
Sivin
Dec 3, 2024
Sivin
32 minutes ago
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Inequalitis
sqing   9
N 2 hours ago by sqing
Let $ a,b,c\geq 0 , a^2+b^2+c^2 =3.$ Prove that
$$a^3 +b^3 +c^3 +\frac{11}{5}abc \leq \frac{26}{5}$$
9 replies
sqing
May 31, 2025
sqing
2 hours ago
J
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Proving Algebra
Justbrick   0
2 hours ago
Prove that for all positive real numbers $x, y, z > 0$ such that $x + y + z = 3$, we have:
\[ \frac{(1 - x)^2}{1 - x^4} + \frac{(1 - y)^2}{1 - y^4} + \frac{(1 - z)^2}{1 - z^4} \ge 0. \]
0 replies
Justbrick
2 hours ago
0 replies
J
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The 24 Game No Postfarming Edition.
maxamc   36
N 2 hours ago by PikaPika999
Use the numbers $1,2,3,4,5,6,7,8,9,10$ (you must use all numbers) and the operations $+,-,\cdot,\div$ ONLY to create all positive integers in order. I will repeat 1 more time: ONLY these 4 operations (no concatenation or anything like totient etc.)

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

Solutions
36 replies
maxamc
Yesterday at 4:10 PM
PikaPika999
2 hours ago
J
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Inequalities
sqing   4
N 2 hours ago by sqing
Let $ a,b,c\geq 0, \frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}=\frac{3}{2}.$ Prove that
$$ \left(a+b+c-\frac{17}{6}\right)^2+9abc \geq\frac{325}{36}$$$$ \left(a+b+c-\frac{5}{2}\right)^2+12abc \geq\frac{49}{4}$$$$\left(a+b+c-\frac{14}{5}\right)^2+\frac{49}{5}abc \geq\frac{49}{5}$$
4 replies
sqing
Jun 30, 2025
sqing
2 hours ago
J
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Inequalities
sqing   11
N 3 hours ago by sqing
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^4}{b^4}+\frac{1}{a^4}+42ab-a^4\geq 43$$$$ \frac{a^5}{b^5}+\frac{1}{a^5}+64ab-a^5\geq 65$$$$ \frac{a^6}{b^6}+\frac{1}{a^6}+90ab-a^6\geq 91$$$$ \frac{a^7}{b^7}+\frac{1}{a^7}+121ab-a^7\geq 122$$
11 replies
sqing
May 28, 2025
sqing
3 hours ago
J
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Math Counts
sudarya   6
N 4 hours ago by DetectiveBanana25
Hello guys!



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


6 replies
sudarya
Yesterday at 3:42 PM
DetectiveBanana25
4 hours ago
J
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9 Pythagorean Triples
ZMB038   85
N 4 hours ago by PikaPika999
Please put some of the ones you know, and try not to troll/start flame wars! Thank you :D
85 replies
ZMB038
May 19, 2025
PikaPika999
4 hours ago
J
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Scores on Contests for Ya'll to submit! (everyone be as honest as possible)
NEILDASEAL_12345   112
N 4 hours ago by DreamineYT
What are yall's most recent scores on mathCON, AMC 8 and other things? I just wanna know how I'm holding up and if my scores are good! This is the end of my first year in contest math, and I've always been gifted, but can someone tell me if these scores are good? MATHCON: 240
AMC 8: 19 QUESTIONS
AMC 10: 16 QUESTIONS
btw I'm going to 7th grade

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

I used formula $A$ $=$ $abc/4R$ in problem 1
to get circumradius as $5$
21 replies
Provyfx
Yesterday at 6:14 PM
JerryZYang
Yesterday at 9:44 PM
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Random but useful theorems
booking   102
N Yesterday at 9:19 PM by Hanruz
There have been all these random but useful theorems
Please post any theorems you know, random or not, but please say whether they are random or not.
I'll start give an example:
Random
I am just looking for some theorems to study.
102 replies
booking
Jul 16, 2025
Hanruz
Yesterday at 9:19 PM
J
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no answer problem
Mathichian   18
N Yesterday at 9:07 PM by codeninja
if you chose a random answer for this problem, what is the probability that you are correct?
A)25% B)50% C)60% D)25%
18 replies
Mathichian
Jul 22, 2025
codeninja
Yesterday at 9:07 PM
J
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9 How many perfect squares do you know?
mymelody1234   9
N Yesterday at 8:37 PM by sadas123
I've memorized 30 perfect squares :)
I meant to say consecutive squares oops
9 replies
mymelody1234
Yesterday at 7:33 PM
sadas123
Yesterday at 8:37 PM
J
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gcd of n(n+1)(n+2)(n+3)(n+4)
Marius_Avion_De_Vanatoare   16
N Yesterday at 8:32 PM by superhuman233
Find the greatest common divisor of the numbers: $1\cdot 2\cdot 3 \cdot 4 \cdot 5$ and $2\cdot 3 \cdot 4 \cdot 5 \cdot 6 \dots 2025 \cdot 2026 \cdot 2027 \cdot 2028 \cdot 2029$.
16 replies
Marius_Avion_De_Vanatoare
Jun 14, 2025
superhuman233
Yesterday at 8:32 PM
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J
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an algebra problem
Asyrafr09   2
N May 4, 2025 by pooh123
Determine all real number($x,y,z$) that satisfy
$$x=1+\sqrt{y-z^2}$$$$y=1+\sqrt{z-x^2}$$$$z=1+\sqrt{x-y^2}$$
2 replies
Asyrafr09
May 4, 2025
pooh123
May 4, 2025
J
an algebra problem
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Reveal topic tags
algebra
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Asyrafr09
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Asyrafr09
#1 h  May 4, 2025, 10:05 AM
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Determine all real number($x,y,z$) that satisfy
$$x=1+\sqrt{y-z^2}$$$$y=1+\sqrt{z-x^2}$$$$z=1+\sqrt{x-y^2}$$
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abartoha
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abartoha
#2 h  May 4, 2025, 10:42 AM
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(1,1,1) is non trivial. Deepseek, GroK says it's the only one.
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pooh123
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pooh123
#3 h  May 4, 2025, 12:03 PM
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Asyrafr09 wrote:
Determine all real number($x,y,z$) that satisfy
$$x=1+\sqrt{y-z^2}$$$$y=1+\sqrt{z-x^2}$$$$z=1+\sqrt{x-y^2}$$

We have the system of equations:
\[ \begin{aligned} x &= 1 + \sqrt{y - z^2} \quad \text{(1)} \\ y &= 1 + \sqrt{z - x^2} \quad \text{(2)} \\ z &= 1 + \sqrt{x - y^2} \quad \text{(3)} \end{aligned} \]
It's easy to see that \(x, y, z \geq 1\).
Without loss of generality, let \(x = \max\{x, y, z\}\).
Then since \(x \geq 1\), we have \(x^2 \geq x \geq z\).
Therefore, from equation (2), \(z - x^2 \leq 0\), so the square root must be zero, i.e. equality must occur.

\[ \Rightarrow z = x^2 \]
Substituting this into (2):

\[ y = 1 + \sqrt{z - x^2} = 1 + \sqrt{0} = 1 \]
Then from (1):

\[ x = 1 + \sqrt{y - z^2} = 1 + \sqrt{1 - 1} = 1 \]
And similarly:

\[ z = 1 + \sqrt{x - y^2} = 1 + \sqrt{1 - 1} = 1 \]
So \(x = y = z = 1\), which is indeed the only solution.
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