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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:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]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.
Posting Guidelines
Update on Basic Forum Rules
What belongs on this forum?
How do I write a thorough solution?
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How do I study for mathcounts?
Mathcounts FAQ and resources
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
prime numbers
wpdnjs   115
N 4 hours ago by SomeonecoolLovesMaths
does anyone know how to quickly identify prime numbers?

thanks.
115 replies
wpdnjs
Oct 2, 2024
SomeonecoolLovesMaths
4 hours ago
MATHCOUNTS
ILOVECATS127   14
N 4 hours ago by ILOVECATS127
Hi,

I am looking to get on my school MATHCOUNTS team next year in 7th grade, and I had a question: Where do the school round questions come from? (Sprint, Chapter, Team, Countdown)
14 replies
ILOVECATS127
Wednesday at 10:56 PM
ILOVECATS127
4 hours ago
Two similar geometry question
ERMSCoach   3
N 4 hours ago by ERMSCoach
I have two questions for you to solve.
However, the particular question I am asking here is
Given the 1st question has answer x, find an easy solution to 2nd question in terms of x.
Essentially are these the same question, or is it a coincidence that the answers are similar?

First question, Equilateral triangle ABC has circle O tangent to BC and passing through A. Circle O intersects AB and AC at D and E, with AD=10 and AE=11. What is the length of BC?

Second question, Quadrilateral ABCD has AB=3, AD=4, B and D are right angles, C is 60 degrees.
What is the length of AC?
3 replies
ERMSCoach
Apr 25, 2025
ERMSCoach
4 hours ago
2025 MATHCOUNTS State Hub
SirAppel   645
N 5 hours ago by jkim0656
Previous Years' "Hubs": (2022) (2023) (2024)Please Read

Now that it's April and we're allowed to discuss ...
[list=disc]
[*] CA: 43 (45 44 43 43 43 42 42 41 41 41)
[*] NJ: 43 (45 44 44 43 39 42 40 40 39 38) *
[*] NY: 42 (43 42 42 42 41 40 38 38 38 38 38 38)
[*] TX: 42 (43 43 43 42 42 40 40 38 38 38)
[*] MA: 41 (45 43 42 41)
[*] WA: 41 (41 45 42 41 41 41 41 41 41 40) *
[*]VA: 40 (41 40 40 40)
[*] FL: 39 (42 41 40 39 38 37 37)
[*] IN: 39 (41 40 40 39 36 35 35 35 34 34)
[*] NC: 39 (42 42 41 39)
[*] IL: 38 (41 40 39 38 38 38)
[*] OR: 38 (44 39 38 38)
[*] PA: 38 (41 40 40 38 38 37 36 36 34 34) *
[*] MD: 37 (43 39 39 37 37 37)
[*] AZ: 36 (40? 39? 39 36)
[*] CT: 36 (44 38 38 36 35 35 34 34 34 33 33 32 32 32 32)
[*] MI: 36 (39 41 41 36 37 37 36 36 36 36) *
[*] MN: 36 (40 36 36 36 35 35 35 34)
[*] CO: 35 (41 37 37 35 35 35 ?? 31 31 30) *
[*] GA: 35 (38 37 36 35 34 34 34 34 34 33)
[*] OH: 35 (41 37 36 35)
[*] AR: 34 (46 45 35 34 33 31 31 31 29 29)
[*] NV: 34 (41 38 ?? 34)
[*] TN: 34 (38 ?? ?? 34)
[*] WI: 34 (40 37 37 34 35 30 28 29 29 29) *
[*] HI: 32 (35 34 32 32)
[*] NH: 31 (42 35 33 31 30)
[*] DE: 30 (34 33 32 30 30 29 28 27 26? 24)
[*] SC: 30 (33 33 31 30)
[*] IA: 29 (33 30 31 29 29 29 29 29 29 29 29 29) *
[*] NE: 28 (34 30 28 28 27 27 26 26 25 25)
[*] SD: 22 (30 29 24 22 22 22 21 21 20 20)
[/list]
Cutoffs Unknown

* means that CDR is official in that state.

Notes

For those asking about the removal of the tiers, I'd like to quote Jason himself:
[quote=peace09]
learn from my mistakes
[/quote]

Help contribute by sharing your state's cutoffs!
645 replies
SirAppel
Apr 1, 2025
jkim0656
5 hours ago
The best math formulas?
anticodon   14
N 6 hours ago by Soupboy0
my math teacher recently offhandedly mentioned in class that "the law of sines is probably in the top 10 of math formulas". This inspired me to make a top 10 list to see if he's right (imo he actually is...)

so I decided, it would be interesting to hear others' opinions on the top 10 and we can compile an overall list.

Attached=my list (sorry if you can't read my handwriting, I was too lazy to do latex, and my normal pencil handwriting looks better)

the formulas
14 replies
anticodon
Yesterday at 11:00 PM
Soupboy0
6 hours ago
sleep tips
Soupboy0   15
N Today at 2:18 AM by giratina3
can someone help me learn how to fall asleep faster bc I'm nervous/excited bc nats is upcoming
15 replies
Soupboy0
Yesterday at 4:20 PM
giratina3
Today at 2:18 AM
9 AMC 10 Prep
bluedino24   32
N Today at 2:17 AM by bluedino24
I'm in 7th grade and thought it would be good to start preparing for the AMC 10. I'm not extremely good at math though.

What are some important topics I should study? Please comment below. Thanks! :D
32 replies
bluedino24
May 2, 2025
bluedino24
Today at 2:17 AM
9 zeroes!.
ericheathclifffry   8
N Today at 2:15 AM by giratina3
i personally have no idea
8 replies
ericheathclifffry
May 5, 2025
giratina3
Today at 2:15 AM
Facts About 2025!
Existing_Human1   262
N Today at 2:14 AM by giratina3
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)
262 replies
Existing_Human1
Jan 1, 2025
giratina3
Today at 2:14 AM
9 What competitions do you do
VivaanKam   23
N Today at 1:24 AM by K124659

I know I missed a lot of other competitions so if you didi one of the just choose "Other".
23 replies
VivaanKam
Apr 30, 2025
K124659
Today at 1:24 AM
MAP Goals
Antoinette14   3
N Today at 1:13 AM by Schintalpati
What's yall's MAP goals for this spring?
Mine's a 300 (trying to beat my brother's record) but since I'm at a 285 rn, 290+ is more reasonable.
3 replies
Antoinette14
Yesterday at 11:59 PM
Schintalpati
Today at 1:13 AM
9 Have you participated in the MATHCOUNTS competition?
aadimathgenius9   53
N Today at 12:19 AM by Math-lover1
Have you participated in the MATHCOUNTS competition before?
53 replies
aadimathgenius9
Jan 1, 2025
Math-lover1
Today at 12:19 AM
9 MathandAI4Girls!!!
Inaaya   13
N Yesterday at 10:28 PM by fossasor
How many problems did y'all solve this year?
I clowned and started the pset the week before :oops:
Though I think if i used the time wisely, I could have at least solved 11 of them
ended up with 9 :wallbash_red:
13 replies
Inaaya
Wednesday at 7:25 PM
fossasor
Yesterday at 10:28 PM
9 What is the best way to learn math???
lovematch13   89
N Yesterday at 9:48 PM by Capybara7017
On the contrary, I'm also gonna try to send this to school admins. PLEASE DO NOT TROLL!!!!
89 replies
lovematch13
May 22, 2023
Capybara7017
Yesterday at 9:48 PM
random problem i just thought about one day
ceilingfan404   27
N Apr 29, 2025 by PikaPika999
i don't even know if this is solvable
Prove that there are finite/infinite powers of 2 where all the digits are also powers of 2. (For example, $4$ and $128$ are numbers that work, but $64$ and $1024$ don't work.)
27 replies
ceilingfan404
Apr 20, 2025
PikaPika999
Apr 29, 2025
random problem i just thought about one day
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ceilingfan404
1135 posts
#1 • 4 Y
Y by aidan0626, e_is_2.71828, Exponent11, PikaPika999
i don't even know if this is solvable
Prove that there are finite/infinite powers of 2 where all the digits are also powers of 2. (For example, $4$ and $128$ are numbers that work, but $64$ and $1024$ don't work.)
This post has been edited 1 time. Last edited by ceilingfan404, Apr 20, 2025, 7:55 PM
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huajun78
71 posts
#2 • 1 Y
Y by PikaPika999
well as the number gets bigger, there are more digits, so it's less likely that ALL the digits will be a power of 2 (1, 2, 4, 8).

for the first 20 powers of 2 after $2^{10}$ ($2^{11}$ to $2^{30}$), none of them satisfy the condition (I tested all of them), so it's very unlikely that numbers with even more digits will.

I don't know how to prove this but that fact suggests that there are only a finite number.
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vincentwant
1385 posts
#3 • 1 Y
Y by PikaPika999
if the number is greater than 512 then the last four digits must be 2112, 4112, 8112, 2224, 4224, 8224, 1424, 1824, 2144, 4144, 8144, 1184, 2128, 4128, 8128, 1248, 2448, 4448, 8448, 2848, 4848, 8848, 2288, 4288, 8288, 1488, 1888

dont think this helps
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Yummo
298 posts
#4 • 1 Y
Y by PikaPika999
@above, what about 1024?
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vincentwant
1385 posts
#5 • 1 Y
Y by PikaPika999
Yummo wrote:
@above, what about 1024?

0 is not a power of 2
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e_is_2.71828
222 posts
#6 • 1 Y
Y by PikaPika999
ceilingfan404 wrote:
i don't even know if this is solvable
Prove that there are finite/infinite powers of 2 where all the digits are also powers of 2. (For example, $4$ and $128$ are numbers that work, but $64$ and $1024$ don't work.)

I won't look into it completely, but we can start somewhere. We'll see if it is possible to "generate" a formula for these numbers. So let $n$ be a $k$-digit number such that $n=a_ka_{k-1}...a_2a_1a_0$. Then $n=10^ka_k+10^{k-1}a_k-1...+10a_1+a_0$, and note for all $i$ $a_i=2^b$, for some $b$. So, $n=10^k \cdot 2^{b_k}+10^{k-1}\cdot 2^{b_{k-1}}+...+10\cdot 2^{b_1}+2^{b_0}$. From there we need also $n=2^c$ for some $c$, and presumably we can take the largest $b_i$, factor it out, and we need the remaining sum to also be a power of $2$. Someone can try working it out from here, I think I started it off well enough.
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wangzrpi
159 posts
#7
Y by
See
https://math.stackexchange.com/questions/2238383/how-many-powers-of-2-have-only-0-or-powers-of-2-as-digits
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e_is_2.71828
222 posts
#8
Y by
Definitely not middle school math
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e_is_2.71828
222 posts
#10 • 1 Y
Y by PikaPika999
K1mchi_ wrote:
e_is_2.71828 wrote:
Definitely not middle school math

its fine

doesn’t need to be msm curriculum
just for msm

if u can’t do it skill issue


"This problem is unlikely to have a simple proof, because the following holds:

Theorem. For any k, there exists a power of 2 whose first k digits and last k digits are all either 1 or 2.
Proof. We begin with looking at the last digits, taking 2nmod10k. For sufficiently large n, 2n≡0(mod2k). Since 2 is a primitive root modulo 5 and modulo 52, it is a primitive root modulo 5k for any k (Wikipedia), so we can have 2n≡b(mod10k) for any b such that b≡0(mod2k).

This is possible to accomplish with only 1 and 2 as digits. We start with b1=2 for k=1, and extend bk−1≡0(mod2k−1) to bk≡0(mod2k) by the rule:

If bk−1≡0(mod2k), take bk=2⋅10k−1+bk−1.
If bk−1≡2k−1(mod2k), take bk=10k−1+bk−1.
(This works because 10k−1≡2k−1(mod2)k.)

There is a unique sequence of digits ending …211111212122112 that we obtain in this way; reversed, it is A023396 in the OEIS.

To make sure that 2n ends in bk, there will be some condition along the lines of
n≡c(modϕ(5k))
or n=c+n′ϕ(5k) for some n′. From there, getting the first k digits to be 1 or 2 is easy along the lines of a recently popular question. We might as well aim for the sequence 111…111k, because we can. To do this, we want
log101.11…1<{(c+n′⋅ϕ(5k))log102}<log101.11…2
where {x} denotes the fractional part of x. This translates into a condition of the form
{n⋅log102ϕ(5k)}∈Ik
for some interval Ik, which we know is possible because α=log102ϕ(5k) is irrational, and therefore the sequence {α},{2α},{3α},… is dense in [0,1].

This concludes the proof.

Instead of the digits {1,2} we could have used the digits {1,4} or {1,8} and given a similar proof; if we multiply the solution to one of these by 2 or 4, we get a power of 2 whose first and last digits come from the set {2,4} or {2,8} or {4,8}. (We can't do this with just the set {0,1} or {0,2} or {0,4} or {0,8}, because eventually we can rule these out by a modular condition.)

It's of course still almost certain that there's no large power of 2 entirely made from the digits {0,1,2,4,8}, but you'd have to say something about the "middle digits" of such a power, which is much harder."

From the stack exchange.
This post has been edited 1 time. Last edited by e_is_2.71828, Apr 24, 2025, 6:01 PM
Reason: Added
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Andrew2019
2310 posts
#12 • 2 Y
Y by e_is_2.71828, Demetri
K1mchi_ wrote:
e_is_2.71828 wrote:
Definitely not middle school math

its fine

doesn’t need to be msm curriculum
just for msm

if u can’t do it skill issue

it would be crazy if someone who has only done the amc 8 and sold on it says others have a skill issue
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maromex
184 posts
#13
Y by
There is a related question: Does the base-$3$ expression of $2^n$ always have a digit equal to $2$ for sufficiently large $n$? If I recall correctly, this problem is unsolved.

The problem discussed in this topic seems similar to this question, and I don't see why it would be solvable with currently known techniques.
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e_is_2.71828
222 posts
#14 • 1 Y
Y by mithu542
I wouldn't listen to someone who can't even spell figure ...
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maxamc
569 posts
#15
Y by
e_is_2.71828 wrote:
I wouldn't listen to someone who can't even spell figure ...

K1mchi_ is always right 100000 aura.
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K1mchi_
85 posts
#16
Y by
Andrew2019 wrote:
K1mchi_ wrote:
e_is_2.71828 wrote:
Definitely not middle school math

its fine

doesn’t need to be msm curriculum
just for msm

if u can’t do it skill issue

it would be crazy if someone who has only done the amc 8 and sold on it says others have a skill issue

slander

i just dont do competitive math

hate me if u like
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MathPerson12321
3749 posts
#17 • 2 Y
Y by e_is_2.71828, mithu542
#11
why dont u?
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K1mchi_
85 posts
#18
Y by
MathPerson12321 wrote:
#11
why dont u?

just quote me


i have better things to do with my time than math rn

i’ll do u the service of enlightenment if i ever find time
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Aaronjudgeisgoat
898 posts
#19
Y by
K1mchi_ wrote:
MathPerson12321 wrote:
#11
why dont u?

just quote me


i have better things to do with my time than math rn

i’ll do u the service of enlightenment if i ever find time

you only have 105 posts, but i feel like ive seen you everywhere
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MathPerson12321
3749 posts
#22
Y by
@bove stop trying to say ur better
do i see mop quals trying to bring me down? no
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RollingPanda4616
255 posts
#24 • 1 Y
Y by PikaPika999
K3mchi_ wrote:
MathPerson12321 wrote:
@bove stop trying to say ur better
do i see mop quals trying to bring me down? no

so? im not trying to bring u down u still bring urself down bc ur very sensitive

dont we celebrate intelligence in our society?

alt alert
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RollingPanda4616
255 posts
#26 • 2 Y
Y by PikaPika999, e_is_2.71828
hey

yes
$~~~~~~$
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valisaxieamc
428 posts
#27 • 3 Y
Y by RollingPanda4616, PikaPika999, e_is_2.71828
Bro imagine making alts cause you fear that aops is going to ban you
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RollingPanda4616
255 posts
#28 • 1 Y
Y by PikaPika999
valisaxieamc wrote:
Bro imagine making alts cause you fear that aops is going to ban you

:rotfl:

anyway let's get this thread back on track

I think you might need to break up the digits and use the prime factorization. (like a 3 digit number $abc$ would be broken down into $a \cdot 2^2 5^2 + b \cdot 2^1 5^1 + c$ and since a, b,c are powers of 2, you could just look at the 5s?) idk how to continue though.
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maromex
184 posts
#29
Y by
I'll say this again:
maromex wrote:
There is a related question: Does the base-$3$ expression of $2^n$ always have a digit equal to $2$ for sufficiently large $n$? If I recall correctly, this problem is unsolved.

The problem discussed in this topic seems similar to this question, and I don't see why it would be solvable with currently known techniques.

Unless a problem about digits has good reason to be solvable with currently known techniques, it's probably not solvable, even if the answer seems obviously true/false at first.

Here's another unsolved problem related to the topic of this thread: For $n > 86$, does $2^n$ always have a $0$ in base $10$?
This post has been edited 1 time. Last edited by maromex, Apr 26, 2025, 7:16 PM
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PikaPika999
1736 posts
#30
Y by
K1mchi_ wrote:
e_is_2.71828 wrote:
Definitely not middle school math

its fine

doesn’t need to be msm curriculum
just for msm

if u can’t do it skill issue

but if the forum is literally called msm, then shouldn't it be msm? plus, if it is harder than msm, there are high school math and college math and high school olympiads, and it could've been placed there?

k1mchi_

not nice
valisaxieamc wrote:
Bro imagine making alts cause you fear that aops is going to ban you

lol imo aops should use ip bans
This post has been edited 1 time. Last edited by PikaPika999, Apr 27, 2025, 11:09 PM
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PikaPika999
1736 posts
#31 • 2 Y
Y by RollingPanda4616, Pengu14
K3mchi_ wrote:
MathPerson12321 wrote:
@bove stop trying to say ur better
do i see mop quals trying to bring me down? no

so? im not trying to bring u down u still bring urself down bc ur very sensitive

dont we celebrate intelligence in our society?

1. True intelligence shines through clarity and simplicity, not overcomplication.
2. Intelligence isn’t just about flaunting knowledge—it’s also about understanding, humility, and connection.
3. True intelligence lies not in power over others, but in empowering those around us.
4. Creativity/intelligence isn’t just about thinking outside the box—it’s about reshaping the box entirely.
5. Leadership isn’t a title—it’s the trust you earn and the influence you wield wisely.
6. Intelligence is not in the answers we give, but in the questions we dare to ask.
7. Intelligence grows when we challenge our own assumptions, not just those of others.
8. The hallmark of intelligence is recognizing that there’s always more to learn.
9. Intelligence flourishes in collaboration, not isolation.
This post has been edited 2 times. Last edited by PikaPika999, Apr 27, 2025, 11:15 PM
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valisaxieamc
428 posts
#33 • 1 Y
Y by PikaPika999
I completely agree with PikaPika but like RollingPanda said, we probably should get back on topic. I mean the kimchi dude is finally leaving us alone and hopefully getting a life so I'll take it as a win
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fake123
93 posts
#34
Y by
bro why are you guys raging over some random kid why can't you just ignore him
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PikaPika999
1736 posts
#35
Y by
fake123 wrote:
bro why are you guys raging over some random kid why can't you just ignore him

we're not raging over "some random kid" who can be ignored (sorry if this sounds harsher than it is)

they start flamewars on multiple different threads. This is how my 1000th post thread got locked :furious

also, they created multiple different alts, which is explicitly said to be against the rules (probably because of getting postbanned from this sheriff

sry if this sounds harsher than i meant to be
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