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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
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[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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
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
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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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
The return of an inequality
giangtruong13   3
N 44 minutes ago by MathsII-enjoy
Let $a,b,c$ be real positive number satisfy that: $a+b+c=1$. Prove that: $$\sum_{cyc} \frac{a}{b^2+c^2} \geq \frac{3}{2}$$
3 replies
giangtruong13
Mar 18, 2025
MathsII-enjoy
44 minutes ago
Inequalities
sqing   7
N an hour ago by sqing
Let $ a,b,c>0 $ and $ a+b\leq 16abc. $ Prove that
$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$Where $ k>0. $
$$ a+b+3c^3\geq\sqrt{\frac{2} {3}}$$$$ a+b+2c^4\geq \frac{5} {8}$$
7 replies
sqing
Yesterday at 12:46 PM
sqing
an hour ago
Inequalities
sqing   10
N an hour ago by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$  (1-abc) (1-a)(1-b)(1-c)  \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c)  \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c)  \le -5840 $$
10 replies
sqing
Jul 12, 2024
sqing
an hour ago
China MO 1996 p1
math_gold_medalist28   1
N an hour ago by MathsII-enjoy
Let ABC be a triangle with orthocentre H. The tangent lines from A to the circle with diameter BC touch this circle at P and Q. Prove that H, P and Q are collinear.
1 reply
math_gold_medalist28
May 2, 2025
MathsII-enjoy
an hour ago
Interesting Diophantine equation
Ro.Is.Te.   2
N Today at 2:22 AM by UberPiggy
Find x, y, z that are positive integers which satisfy this equation $$\frac{1}{x+1} + \frac{1}{y+2} + \frac{1}{z+3} = \frac{11}{12}$$
2 replies
Ro.Is.Te.
Yesterday at 11:26 AM
UberPiggy
Today at 2:22 AM
prime numbers
wpdnjs   103
N Today at 1:05 AM by megarnie
does anyone know how to quickly identify prime numbers?

thanks.
103 replies
wpdnjs
Oct 2, 2024
megarnie
Today at 1:05 AM
9 Am I going insane?
PenguFish   19
N Today at 12:15 AM by nmlikesmath
I feel stupid honestly, and fyi, I did go over every question. Focus is counting with symmetry
19 replies
PenguFish
Mar 26, 2025
nmlikesmath
Today at 12:15 AM
How to get a 300+ on the NWEA MAP MATH test (URGENT)
nmlikesmath   14
N Yesterday at 11:22 PM by nmlikesmath
I have 4 days till this test, I'm wondering how do I get a 300+ and what do I need to know, thank you.
14 replies
nmlikesmath
May 3, 2025
nmlikesmath
Yesterday at 11:22 PM
Floor function
Ro.Is.Te.   5
N Yesterday at 10:55 PM by Soupboy0
Find all the real solution for this equation $$\left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{2x}{3} \right\rfloor = x$$
5 replies
Ro.Is.Te.
May 3, 2025
Soupboy0
Yesterday at 10:55 PM
National Team Round Problem 10?!
Leeoz   5
N Yesterday at 10:53 PM by Soupboy0
This was the 2015 national team round problem 10...


[quote=The hardest problem in MathCounts]
In the city of Trichotomy, every day the weather is exactly one of the following:
sunny, cloudy or rainy. Each day has a 50% chance of having the same weather
as the day before and a 25% chance of having each of the other two types of
weather. If it does not rain on Friday, what is the probability that there will be
no rain during the weekend (Saturday and Sunday)? Express your answer as a
common fraction.
[/quote]
5 replies
Leeoz
Yesterday at 5:44 AM
Soupboy0
Yesterday at 10:53 PM
9 Have you participated in the MATHCOUNTS competition?
aadimathgenius9   39
N Yesterday at 10:41 PM by pingpongmerrily
Have you participated in the MATHCOUNTS competition before?
39 replies
aadimathgenius9
Jan 1, 2025
pingpongmerrily
Yesterday at 10:41 PM
FTW Rating
ScrappyBoss0825   1
N Yesterday at 9:25 PM by sanaops9
Does anyone know when you get your FTW rating?? please tell me.
1 reply
ScrappyBoss0825
Yesterday at 9:17 PM
sanaops9
Yesterday at 9:25 PM
Facts About 2025!
Existing_Human1   259
N Yesterday at 8:49 PM by Math-lover1
Hello AOPS,

As we enter the New Year, the most exciting part is figuring out the mathematical connections to the number we have now temporally entered

Here are some facts about 2025:
$$2025 = 45^2 = (20+25)(20+25)$$$$2025 = 1^3 + 2^3 +3^3 + 4^3 +5^3 +6^3 + 7^3 +8^3 +9^3 = (1+2+3+4+5+6+7+8+9)^2 = {10 \choose 2}^2$$
If anyone has any more facts about 2025, enlighted the world with a new appreciation for the year


(I got some of the facts from this video)
259 replies
Existing_Human1
Jan 1, 2025
Math-lover1
Yesterday at 8:49 PM
300 MAP Goal??
Antoinette14   77
N Yesterday at 7:50 PM by HoneyHap
Hey, so as a 6th grader, my big goal for MAP this spring is to get a 300 (ambitious, i know). I'm currently at a 285 (288 last year though). I'm already taking a intro to counting and probability course (One of my weak points), but is there anything else you recommend I focus on to get a 300?
77 replies
Antoinette14
Jan 30, 2025
HoneyHap
Yesterday at 7:50 PM
Combinatorial proof
MathBot101101   11
N Apr 24, 2025 by MathBot101101
Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}=1-\frac{1}{{n+1)!}
without induction and using only combinatorial arguments?

Induction proof wasn't quite as pleasing for me.
11 replies
MathBot101101
Apr 20, 2025
MathBot101101
Apr 24, 2025
Combinatorial proof
G H J
G H BBookmark kLocked kLocked NReply
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MathBot101101
17 posts
#1
Y by
Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}=1-\frac{1}{{n+1)!}
without induction and using only combinatorial arguments?

Induction proof wasn't quite as pleasing for me.
This post has been edited 1 time. Last edited by MathBot101101, Apr 21, 2025, 6:45 AM
Reason: im dum
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MathBot101101
17 posts
#2
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Bump....
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franklin2013
274 posts
#3
Y by
MathBot101101 wrote:
Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}
without induction and using only combinatorial arguments?

Induction proof wasn't quite as pleasing for me.

$\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}$

what are you trying to prove

EDIT: 250th POST :D
This post has been edited 2 times. Last edited by franklin2013, Apr 20, 2025, 1:42 PM
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Bummer12345
139 posts
#4
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?
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maromex
179 posts
#5 • 1 Y
Y by MathBot101101
I believe the task is \[ \sum\limits_{i=1}^n \dfrac{i}{(i + 1)!} = 1 - \dfrac{1}{(n+1)!}. \]This is indeed a simple induction exercise. As for combinatorial arguments, maybe we can rearrange this into \[  \frac{1}{(n+1)!} + \sum\limits_{i=1}^{n} \frac{i}{(i+1)!} = 1,\]but then I'm not sure how to get a probability that is clearly equal to the LHS and is also equal to $1$.
This post has been edited 1 time. Last edited by maromex, Apr 20, 2025, 8:14 PM
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Kempu33334
609 posts
#6 • 1 Y
Y by MathBot101101
MathBot101101 wrote:
Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}
without induction and using only combinatorial arguments?

Induction proof wasn't quite as pleasing for me.

This isn't an answer, but I think induction is overkill, since it seems you are looking for an intuitive proof, rather than induction. You could use telescoping:

We have that $\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}$ is equal to something (let's call $S$). Adding $\frac{1}{(n+1)!}$ gives that
\begin{align*}
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}+\frac{1}{(n+1)!} &= S+\frac{1}{(n+1)!} \\
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n-1}{n!}+\frac{1}{n!} &= S+\frac{1}{(n+1)!} \\ 
\vdots \\
\frac{1}{2!}+\frac{1}{2!} &= S+\frac{1}{(n+1)!} \\ 
1 &= S+\frac{1}{(n+1)!} \\ 
1-\frac{1}{(n+1)!} &= S.
\end{align*}(This is induction at heart.)
This post has been edited 3 times. Last edited by Kempu33334, Apr 20, 2025, 9:58 PM
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MathBot101101
17 posts
#7
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Extremely sorry, I forgot to write the entire proof statement. Sorry.
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MathBot101101
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#8
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Thank you @maromex and @Kempu33334
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MathBot101101
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#9
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(this might be unreadable because I can't use Latex yet... aops isn't allowing me)

1-(1/(n-1)!) is just probability that we have a permutation of {1, 2, 3, ..., n+1} (suppose a_1, a_2, a_3, ..., a_n+1) with some k in set {1, 2, 3, ..., n} such that a_k>a_k+1. Only permutations without this is a_i=i.
So, 1-(1/(n-1)!)

I can't calculate alternative method.

(Can someone Latex this? Thank you)
This post has been edited 1 time. Last edited by MathBot101101, Apr 23, 2025, 6:47 AM
Reason: latex this pls
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MathBot101101
17 posts
#10
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Bump.....
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MathBot101101
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#11
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Bumpity Bump Bump....
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MathBot101101
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#12
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bump again ig
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