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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.
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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.
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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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Functional equation
socrates   30
N 6 minutes ago by NicoN9
Source: Baltic Way 2014, Problem 4
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
30 replies
socrates
Nov 11, 2014
NicoN9
6 minutes ago
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Algebra Functions
pear333   1
N 16 minutes ago by whwlqkd
Let $P(z)=z-1/z$. Prove that there does not exist a pair of rational numbers $x,y$ such that $P(x)+P(y)=4$.
1 reply
pear333
Today at 12:20 AM
whwlqkd
16 minutes ago
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Polynomial equation with rational numbers
Miquel-point   2
N 20 minutes ago by Assassino9931
Source: Romanian TST 1979 day 2 P1
Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
Dumitru Bușneag
2 replies
Miquel-point
Apr 15, 2023
Assassino9931
20 minutes ago
J
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Area problem
MTA_2024   0
22 minutes ago
Let $\omega$ be a circle inscribed inside a rhombus $ABCD$. Let $P$ and $Q$ be variable points on $AB$ and $AD$ respectively, such as $PQ$ is always the tangent line to $\omega$.
Prove that for any position of $P$ and $Q$ the area of triangle $\triangle CPQ$ is the same.
0 replies
MTA_2024
22 minutes ago
0 replies
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Cutting Cheese
mithu542   5
N Today at 1:25 AM by ohiorizzler1434
A cube of cheese $C=\{(x,y,z)|0\le x,y,z\le 1\}$ is cut along the planes $x=y, y=z$, and $z=x$. How many pieces are there? (No cheese is moved until all three cuts are made.)

sol confirmation
5 replies
mithu542
Feb 28, 2025
ohiorizzler1434
Today at 1:25 AM
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cool math problem
Soupboy0   3
N Yesterday at 11:56 PM by ohiorizzler1434
Call a number skibidi if it can be expressed as $3k+1$, where $k$ is a nonnegative integer. Call a number gyatt if it has $5$ or $7$ factors. What is the sum of all skibidi numbers less than $1000$ that are also gyatt numbers?


answer confirmation
3 replies
Soupboy0
Yesterday at 11:30 PM
ohiorizzler1434
Yesterday at 11:56 PM
J
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Counting problem
Math-lover1   3
N Yesterday at 10:13 PM by sadas123
There's probably a faster way than just listing out all the cases (I hope)
3 replies
Math-lover1
Yesterday at 9:45 PM
sadas123
Yesterday at 10:13 PM
J
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2014 National MathCounts Problem 30
ilikemath247365   6
N Yesterday at 8:24 PM by ibmo0907
The area of the largest equilateral triangle that can be inscribed in a square of side length $1$ unit can be expressed in the form $a\sqrt{b} - c$ units$^{2}$, where $a, b,$ and $c$ are integers. What is the value of $a + b + c$?
6 replies
ilikemath247365
Yesterday at 7:35 PM
ibmo0907
Yesterday at 8:24 PM
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2014 National MathCounts Problem 28
ilikemath247365   5
N Yesterday at 8:00 PM by ilikemath247365
If $f(x) = \frac{ax + b}{cx + d}, {abcd \neq 0}$ and $f(f(x)) = x$ for all $x$ in the domain of $f$, what is the value of $a + d$?
5 replies
ilikemath247365
Yesterday at 7:13 PM
ilikemath247365
Yesterday at 8:00 PM
J
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amc8 2025 difficulty ranking
Soupboy0   18
N Yesterday at 6:14 PM by Catcumber
My opinion of this years amc8 (easiest to hardest)

2 1 8 4 3 6 9 5 11 12 14 10 7 16 17 13 20 15 18 23 19 24 25 21 22
18 replies
Soupboy0
Yesterday at 4:20 PM
Catcumber
Yesterday at 6:14 PM
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k PI DAY! πππ
FiestyTiger82   68
N Yesterday at 5:29 PM by srsh06
THE OFFICIAL AOPS HOLIDAY! πππππππππππππ
I'm sure there are other threads, but let's see how many pi posts this can get before pi day is out.
GO GO GO!

(no spamming)
68 replies
FiestyTiger82
Friday at 8:54 PM
srsh06
Yesterday at 5:29 PM
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Math Problem Level
NS0004   5
N Yesterday at 5:27 PM by sadas123
So I came up with these problems and was wondering what problem number they would be on an AMC 10?

Problem 1:
Triangle ABC is circumscribed by circle T. If T lies on the longest side of Triangle ABC, the side lengths of ABC can be represented as (x-5), (1.5x+6), and (2x-1). Sphere V is then constructed with a radius of (x/2). What is the the volume of sphere V?


Problem 2: Amy is thinking of a number n. n is a 4 digit number, whose 1st and 3rd digit are 2, and can be represented as x^2 with x's first digit as 4. If n's prime factorization is written as a^b x c^d then what is
((a x b x c x d)/5!)?

Sorry for no latex...
These problems are assuming that the answer choices are good.
5 replies
NS0004
Yesterday at 4:45 PM
sadas123
Yesterday at 5:27 PM
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My MATHCOUNTS journey + I need advice
nitride   34
N Yesterday at 5:20 PM by MathPerson12321
So I don't really care if I dox myself but I went to one of the single worst chapters in all of Florida, the Indian river chapter(even the runner of mathcounts states was sayin we would lose to osprey in states). I don't exactly know what my score is, but all I know is that I got 1st place sprint and chapter but then sold CD. If I had to guess, my score was around the 30-37 range. I also mocked at around a 70 on amc 10. My questions are:
1.
What are my chances of me getting nats qual
2.
Should I study the day before the competition
34 replies
nitride
Mar 13, 2025
MathPerson12321
Yesterday at 5:20 PM
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MATHCOUNTS Chapter Score Thread
apex304   101
N Yesterday at 2:30 PM by sadas123
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ \end{tabular}$
101 replies
apex304
Mar 1, 2025
sadas123
Yesterday at 2:30 PM
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Irrationals and powers
Rukevwe   1
N Nov 16, 2022 by iammocnhan
Source: 2016 Nigerian Senior MO Round 2
Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.
1 reply
Rukevwe
Nov 16, 2022
iammocnhan
Nov 16, 2022
J
Irrationals and powers
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Reveal topic tags
induction
Irrational numbers
algebra
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G H BBookmark kLocked kLocked NReply
Source: 2016 Nigerian Senior MO Round 2
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Rukevwe
288 posts
Rukevwe
#1 h  Nov 16, 2022, 10:02 AM • 1 Y
Y by Mango247
Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.
Z K Y
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iammocnhan
403 posts
iammocnhan
#2 h  Nov 16, 2022, 1:22 PM
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Applying binomial expansion we have: $(2+\sqrt{3})^{2n}=\displaystyle \sum _{i=0}^{2n} \left[\binom{2n}{i} 2^{i}\cdot (\sqrt{3})^{2n-i}\right]=\displaystyle \sum _{i=0}^{n} \left[\binom{2n}{2i} 2^{2i}\cdot 3^{n-i}\right]+\displaystyle \sum _{i=1}^{n} \left[\binom{2n}{2i-1} 2^{2i-1}\cdot \sqrt{3} \cdot 3^{n-i}\right]$
$(2-\sqrt{3})^{2n}=\displaystyle \sum _{i=0}^{2n} \left[\binom{2n}{i} 2^{i}\cdot (-\sqrt{3})^{2n-i}\right]=\displaystyle \sum _{i=0}^{n} \left[\binom{2n}{2i} 2^{2i}\cdot 3^{n-i}\right]-\displaystyle \sum _{i=1}^{n} \left[\binom{2n}{2i-1} 2^{2i-1}\cdot \sqrt{3} \cdot 3^{n-i}\right]$
Therefore we have $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}=2\displaystyle \sum _{i=0}^{n} \left[\binom{2n}{2i} 2^{2i}\cdot 3^{n-i}\right]$ is an even integer.
$(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=2\displaystyle \sum _{i=1}^{n} \left[\binom{2n}{2i-1} 2^{2i-1}\cdot \sqrt{3} \cdot 3^{n-i}\right]=2\sqrt{3}\displaystyle \sum _{i=1}^{n} \left[\binom{2n}{2i-1} 2^{2i-1}\cdot 3^{n-i}\right]$
This post has been edited 1 time. Last edited by iammocnhan, Nov 16, 2022, 1:24 PM
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