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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.
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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?
How do I write a thorough solution?
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Mathcounts and how to learn

As always, if you have any questions, you can PM me or any of the other Middle School Moderators. Once again, if you see spam, it would help a lot if you filed a report instead of responding :)

Marathons!
Relays might be a better way to describe it, but these threads definitely go the distance! One person starts off by posting a problem, and the next person comes up with a solution and a new problem for another user to solve. Here's some of the frequently active marathons running in this forum:
[list][*]Algebra
[*]Prealgebra
[*]Proofs
[*]Factoring
[*]Geometry
[*]Counting & Probability
[*]Number Theory[/list]
Some of these haven't received attention in a while, but these are the main ones for their respective subjects. Rather than starting a new marathon, please give the existing ones a shot first.

You can also view marathons via the Marathon tag.

Think this list is incomplete or needs changes? Let the mods know and we'll take a look.
368 replies
jellymoop
May 8, 2015
harry1234
May 16, 2018
J
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Simple vector geometry existence
AndreiVila   3
N 7 minutes ago by Ianis
Source: Romanian District Olympiad 2025 9.1
Let $ABCD$ be a parallelogram of center $O$. Prove that for any point $M\in (AB)$, there exist unique points $N\in (OC)$ and $P\in (OD)$ such that $O$ is the center of mass of $\triangle MNP$.
3 replies
AndreiVila
Mar 8, 2025
Ianis
7 minutes ago
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Checkerboard
Ecrin_eren   1
N 21 minutes ago by Ecrin_eren
On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)
1 reply
1 viewing
Ecrin_eren
Today at 5:20 AM
Ecrin_eren
21 minutes ago
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BD tangent to (MDE) , rhombus ABCD with <DCB=60^o
parmenides51   1
N 44 minutes ago by vanstraelen
Source: 2021 Germany R4 10.6 https://artofproblemsolving.com/community/c3208025_
Let a rhombus $ABCD$ with $|\angle DCB| = 60^o$ be given . On the extension of the segment $\overline{CD}$ beyond $D$, a point $E$ is chosen arbitrarily. Let the line through $E$ and $A$ intersect the line $BC$ at the point $F$. Let $M$ be the intersection of the lines $BE$ and $DF$. Prove that the line $BD$ is tangent to the circumcircle of the triangle $MDE$.
1 reply
parmenides51
Oct 6, 2024
vanstraelen
44 minutes ago
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Geometry Problem #42
vankhea   2
N an hour ago by kaede_Arcadia
Source: Van Khea
Let $P$ be any point. Let $D, E, F$ be projection point from $P$ to $BC, CA, AB$. Circumcircle $(ABC)$ cuts circumcircle $(AEF), (BFD), (CDE)$ at $A_1, B_1, C_1$. Let $A_2, B_2, C_2$ be antipode of $A_1, B_1, C_1$ wrt $(AEF), (BFD), (CDE)$. Prove that $A_2, B_2, C_2, P$ are cyclic.
2 replies
vankhea
Sep 6, 2023
kaede_Arcadia
an hour ago
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Basic Maths
JetFire008   6
N 4 hours ago by mathelvin
Find $x$: $\sqrt{9}x=18$
6 replies
JetFire008
Today at 1:19 PM
mathelvin
4 hours ago
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MATHCOUNTS Chapter Score Thread
apex304   107
N 4 hours ago by alwaysgonnagiveyouup
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ \end{tabular}$
107 replies
apex304
Mar 1, 2025
alwaysgonnagiveyouup
4 hours ago
J
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Really Nasty MathCounts Problem
ilikemath247365   16
N 5 hours ago by AtlantisII
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?
16 replies
ilikemath247365
Mar 14, 2025
AtlantisII
5 hours ago
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state mathcounts colorado
aoh11   60
N 5 hours ago by sadas123
I have state mathcounts tomorrow. What should I do to get prepared btw, and what are some tips for doing sprint and cdr?
60 replies
aoh11
Mar 15, 2025
sadas123
5 hours ago
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How important is math "intuition"
Dream9   16
N 5 hours ago by Dream9
When I see problems now, they usually fall under 3 categories: easy, annoying, and cannot solve. Over time, more problems become easy, but I don't think I'm learning anything "new" so is higher level math like AMC 10 more about practice, so you know what to do when you see a problem? Of course, there's formulas for some problems but when reading a lot of solutions I didn't see many weird formulas being used and it was just the way to solve the problem was "odd".
16 replies
Dream9
Mar 19, 2025
Dream9
5 hours ago
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Chances of mathcounts nats qual
stjwyl   84
N 5 hours ago by AtlantisII
Info:
In 8th grade so I'm really hoping I can make nats now

I currently mock around 38 - 40 on nationals questions from 2015+
I mock anywhere from 37 - 42 on state questons from 2020+

For the sprint round I also have noticed that the difficulty jump from questions around 19 and 20 to questions around 22 and 23 has been really large (starting from 2023). I've also noticed that the last three questions (also from 2023 ->) are IMO impossible to do in the 40 minutes.

On target I can get 7/8 or even 8/8 if I'm lucky but it's possible for me to get 6/8

I'm in MA :sob: really hard state so do I have a chance

Edit: Just mocked the 2022 state round and got a 41 (29 sprint, 12 target :sob:)

Currently putting around 3 hrs or so a day and I have been for the past 2 months
States is 3/1 for me :sob:

so am i cooked
84 replies
stjwyl
Feb 21, 2025
AtlantisII
5 hours ago
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Three variable equations
gpen1000   1
N 5 hours ago by fruitmonster97
1. For integers $x$, $y$, and $z$ such that $\frac{\sqrt{x}}{\sqrt{y}} = z$, find $\frac{\sqrt{z}}{\sqrt{x}}$ in terms of $x$, $y$, and $z$.

2. For integers $x$, $y$, and $z$ such that $x = y - 1$ and $z = y + 1$, prove that $y^3 = xyz + y$.
1 reply
gpen1000
Today at 1:55 PM
fruitmonster97
5 hours ago
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Why was this poll blocked
jkim0656   11
N 5 hours ago by skronkmonster
Hey AoPS ppl!
I made a poll about Pi vs Tau over here:
https://artofproblemsolving.com/community/c3h3527460
But after a few days it got blocked but i don't get why?
how is this harmful or different from other polls?
It really wasn't that harmful or popular i got to say tho... :noo:
11 replies
jkim0656
Mar 18, 2025
skronkmonster
5 hours ago
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MC nationals 2023 sprint Q28
Soupboy0   5
N 6 hours ago by Bummer12345
What common fraction is equivalent to the expression shown?

$\frac{\frac{1}{2021 \cdot 2022} + \frac{2}{2019 \cdot 2021} + \frac{3}{2017 \cdot 2020} + ... + \frac{1010}{3 \cdot 1013} + \frac{1011}{1 \cdot 1012}}{\frac{2022}{1011}+\frac{2022}{1010}+\frac{2022}{1009}+...+\frac{2022}{2}+\frac{2022}{1}}$?

can someone please help me
5 replies
Soupboy0
Mar 12, 2025
Bummer12345
6 hours ago
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Mathcounts STRATEGIES
Existing_Human1   17
N 6 hours ago by Soupboy0
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
17 replies
Existing_Human1
Yesterday at 7:27 PM
Soupboy0
6 hours ago
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nice problem
Mathuzb   5
N May 24, 2019 by Mathuzb
Let $a,b,c>0$. Prove that $a^2+b^2+c^2 \geq a(\frac{c^3+b^3}{2})^{\frac{1}{3}}+b(\frac{a^3+c^3}{2})^{\frac{1}{3}}+ c(\frac{a^3+b^3}{2})^{\frac{1}{3}}$
5 replies
Mathuzb
May 24, 2019
Mathuzb
May 24, 2019
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nice problem
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Reveal topic tags
inequalities
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Mathuzb
431 posts
Mathuzb
#1 h  May 24, 2019, 12:44 PM • 2 Y
Y by Adventure10, Mango247
Let $a,b,c>0$. Prove that $a^2+b^2+c^2 \geq a(\frac{c^3+b^3}{2})^{\frac{1}{3}}+b(\frac{a^3+c^3}{2})^{\frac{1}{3}}+ c(\frac{a^3+b^3}{2})^{\frac{1}{3}}$
This post has been edited 2 times. Last edited by Mathuzb, May 24, 2019, 12:44 PM
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Math-wiz
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#2 h  May 24, 2019, 12:45 PM • 2 Y
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Power mean inequality maybe.
@below trying. I was just giving hint
This post has been edited 1 time. Last edited by Math-wiz, May 24, 2019, 12:48 PM
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Mathuzb
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#3 h  May 24, 2019, 12:46 PM • 2 Y
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Math-wiz wrote:
Power mean inequality maybe

Ok, can you use?
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NaPrai
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#4 h  May 24, 2019, 1:52 PM • 4 Y
Y by teomihai, Mathuzb, Adventure10, Mango247
By AM-GM, we have $$\sum_{cyc}a\sqrt[3]{\frac{b^3+c^3}{2}} = \sum_{cyc} \sqrt[3]{a^2(b^2+c^2-bc)\left(\frac{ab+ac}{2}\right)} \le \sum_{cyc}\frac{a^2+b^2+c^2-bc+\frac{ab+ac}{2}}{3} = \sum_{cyc} a^2.$$
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arqady
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#5 h  May 24, 2019, 1:55 PM • 5 Y
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Use $$\frac{a^2+b^2}{a+b}\geq\sqrt[3]{\frac{a^3+b^3}{2}}.$$
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Mathuzb
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#6 h  May 24, 2019, 10:46 PM • 1 Y
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NaPrai wrote:
By AM-GM, we have $$\sum_{cyc}a\sqrt[3]{\frac{b^3+c^3}{2}} = \sum_{cyc} \sqrt[3]{a^2(b^2+c^2-bc)\left(\frac{ab+ac}{2}\right)} \le \sum_{cyc}\frac{a^2+b^2+c^2-bc+\frac{ab+ac}{2}}{3} = \sum_{cyc} a^2.$$

fantastic solution, thank you very much!
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