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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Today at 3:57 PM
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

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0 replies
jlacosta
Today at 3:57 PM
0 replies
J
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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.
Posting Guidelines
Update on Basic Forum Rules
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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Original Problem on Logarithms
yes45   2
N an hour ago by vincentwant
What is the value of $\log _{xyz}{n}$ if
$\log _{xy}{n} = 48$,
$\log _{yz}{n} = 64$, and
$\log _{z}{n} = 96$?

Answer

Solution
2 replies
yes45
Today at 3:09 PM
vincentwant
an hour ago
J
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Share your favorite problem
the_mathtrovert_next_door   8
N 2 hours ago by Yiyj
hi, this is my first post, so consider yourselves warned, here's how this thread works: you send your favorite problem, and send the solutions to the previous problem that was sent, here's mine;

Consider a circle that has 2n points on the circumference that are equally spaced. Alice and Bob are playing a game such that Alice will pick a number d such that d is less than or equal to n. Bob wins if he can colour the 2n points with n colours such that the 2 points of the same colour have exactly d-1 points between them. Devise a winning strategy for Alice.
8 replies
the_mathtrovert_next_door
Today at 4:09 PM
Yiyj
2 hours ago
J
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9 Prodigy AoPS or Khanacadamy
ZMB038   107
N 3 hours ago by GallopingUnicorn45
Hey everyone just was wondering what everybody prefers? Try not to fight so this doesn't get locked!
107 replies
ZMB038
May 22, 2025
GallopingUnicorn45
3 hours ago
J
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MIT PRIMES STEP
pingpongmerrily   6
N 4 hours ago by Multpi12
Anyone else applying? How cooked am I for the placement test... (106.5 AMC 10, 5 AIME, 36/27 States/Nationals)
6 replies
pingpongmerrily
May 30, 2025
Multpi12
4 hours ago
J
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Find the value of pq+rs/ps+qr
Darealzolt   1
N 5 hours ago by vanstraelen
Let \(p,q,r,s\) be real numbers that satisfy
\[ p^2+q^2=r^2+s^2 \]\[ p^2+s^2-ps=q^2+r^2+qr \]Hence find the value of \(\frac{pq+rs}{ps+qr}\)

Click to reveal hidden text
1 reply
Darealzolt
Yesterday at 3:40 AM
vanstraelen
5 hours ago
J
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Original Problem (Complex Numbers)
qrxz17   1
N 6 hours ago by sultanine
Problem: Given that for a complex number \(z\), the value of \(z \cdot \overline{z} +(z+\overline{z})\) is \(19\). Find all complex numbers \(z\).

Answer: Click to reveal hidden text

Solution: Writing the given equation in standard form, we get
\begin{align*} (a+bi)\cdot(a-bi) +[(a+bi)+(a-bi)] =a^2+b^2+2a. \end{align*}
Then we have
\begin{align*} a^2+b^2+2a &= 19 \\ (a+1)^2+b^2 &=20 \end{align*}
This is the standard equation of a circle. So, \(r^2=20\) and \(r=2\sqrt{5}\).

Therefore, the combined value of the sum and product of a complex number and its conjugate is \(19\) when the complex numbers lie on the circle centered at \((-1,0)\) with radius \(2\sqrt{5}\).
1 reply
qrxz17
Today at 3:18 PM
sultanine
6 hours ago
J
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Daily Problem Writing Practice
KSH31415   2
N 6 hours ago by sultanine
I'm trying to get better at writing problems so I decided to challenge myself to write one problem for every day this month (June 2025). I will post them in this thread as well as edit this post with all of them in hide tags. If I can, I'll include a difficulty level in the form of AIME placement. If anybody wants to solve them and give feedback on the problem and/or my difficulty rating, please do!

June 1 (AIME P4)

June 1 (AIME P4)
A bag contains $6$ red balls and $6$ blue balls. A draw consists of randomly selecting $2$ of the remaining balls from bag without replacement, and then setting them aside. A draw is called a match if the two balls have the same color. Compute the expected number of draws until either a match occurs or the bag is empty.
2 replies
KSH31415
Today at 3:13 PM
sultanine
6 hours ago
J
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k Worst Sillies of All Time
pingpongmerrily   75
N Today at 3:34 PM by Aaronjudgeisgoat
Share the worst sillies you have ever made!

Mine was probably on the 2024 MathCounts State Target Round Problem 8, where I wrote my answer as a fraction instead of a percent, which cost me a trip to Nationals that year.
75 replies
pingpongmerrily
May 30, 2025
Aaronjudgeisgoat
Today at 3:34 PM
J
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Phillips Exeter is looking for math kids! That means YOU!
enya_yurself   69
N Today at 2:54 PM by idk12345678
I have received some insider information that may or may not prove to be helpful to you all. I am not sure where the best place to post this is, but somebody recommended msm so here I am.

[quote]admissions was explicitly told to accept more math kids, 25 26 and 27 have been pretty disappointing for the math dept bc of covid[/quote]
(source: a friend who talked to a faculty member on the admissions committee)

This means that Phillips Exeter is looking for more people like you all! I hope y'all choose to apply!

Remember that Exeter offers need blind financial aid :)
69 replies
enya_yurself
Aug 9, 2024
idk12345678
Today at 2:54 PM
J
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Worst math problems
LXC007   57
N Today at 5:30 AM by Andrew2019
What is the most egregiously bad problem or solution you have encountered in school?
57 replies
LXC007
May 21, 2025
Andrew2019
Today at 5:30 AM
J
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9 Square roots
A7456321   22
N Today at 3:22 AM by elizhang101412
Me personally I only have $\sqrt2=1.414$ memorized but I'm sure there are people out there with more!
22 replies
A7456321
May 29, 2025
elizhang101412
Today at 3:22 AM
J
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prime numbers
wpdnjs   130
N Today at 1:57 AM by ayeshaaq
does anyone know how to quickly identify prime numbers?

thanks.
130 replies
wpdnjs
Oct 2, 2024
ayeshaaq
Today at 1:57 AM
J
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1434th post
vincentwant   22
N Today at 1:41 AM by vincentwant
This is my 1434th post. Here are some of my favorite (non-1434-related) problems that I wrote for various contests over the past few years. A $\star$ indicates my favorites.

-----

A function $f(x)$ is defined over the positive integers as follows: $f(1)=0$, $f(p^n)=n$ for $p$ prime, and for all relatively prime positive integers $a$ and $b$, $f(ab)=f(a)f(b)+f(a)+f(b)$. If $N$ is the smallest positive integer such that $f(N)=20$, find the units digit of $N$.

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~6 \qquad\textbf{(E)} ~8$

(2023 VMAMC 10 #23)

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$\star$ If convex quadrilateral $ABCD$ satisfies $AB=6$, $\angle CAB=30^{\circ}$, $\angle CDB=60^{\circ}$, $\angle BCD-\angle ABC=30^{\circ}$, and $CD=1$, what is the value of $BC^2$? Express your answer in simplest radical form.

(2024 STNUOCHTAM Sprint #30)

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Let $N=1!\cdot2!\cdot4!\cdot8!\cdots(2^{1000})!$ and $d$ be the greatest odd divisor of $N$. Let $f(n)$ for even $n$ denote the product of every odd positive integer less than $n$. If $d=f(a_1)^{b_1}f(a_2)^{b_2}f(a_3)^{b_3}\cdots f(a_k)^{b_k}$ for positive integers $a_1,a_2,\dots,a_k$ and $b_1,b_2,\dots,b_k$ where $k$ is minimized, find the number of divisors of $a_1a_2a_3\cdots a_k{}$.

(2024 STNUOCHTAM Sprint #29)

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$\star$ There exists exactly one positive real number $k$ such that the graph of the equation $\frac{x^3+y^3}{xy-k}=k$ consists of a line and a point not on the line. The distance from the point to the line can be expressed as $\frac{a}{\sqrt{b}}$, where $a$ and $b$ are positive integers and $b$ is not divisible by any square greater than $1$. Find $a+b$.

(2023-2024 WOOT AIME 3 #12)

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Let $a\odot b=\frac{4ab}{a+b+2\sqrt{ab}}$. If $x,y,z,k$ are positive real numbers such that $x\odot y=4z$, $x\odot z=\frac{9}{4}y$, and $y\odot z=kx$, find $k$. Express your answer as a common fraction.

(2024 STNUOCHTAM Sprint #26)

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$\star$ Let $ABCDE$ be a convex pentagon satisfying $AB = BC = CD = DE$, $\angle ABC = \angle CDE$, $\angle EAB = \angle AED = \frac{1}{2}\angle BCD$. Let $X$ be the intersection of lines $AB$ and $CD$. If $\triangle BCX$ has a perimeter of $18$ and an area of $11$, find the area of $ABCDE$.

$\textbf{(A)} ~136 \qquad\textbf{(B)} ~137 \qquad\textbf{(C)} ~138 \qquad\textbf{(D)} ~139 \qquad\textbf{(E)} ~140$

(2024 TMC AMC 10 #25)

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$\star$ Let $\triangle{ABC}$ be an acute scalene triangle with longest side $AC$. Let $O$ be the circumcenter of $\triangle{ABC}$. Points $X$ and $Y$ are chosen on $AC$ such that $OX\perp BC$ and $OY\perp AB$. If $AX=7$, $XY=8$, and $YC=9$, the area of the circumcircle of $\triangle{ABC}$ can be expressed as $k\pi$. Find $k$.

$\textbf{(A)} ~145\qquad \textbf{(B)} ~148\qquad \textbf{(C)} ~153\qquad \textbf{(D)} ~157\qquad \textbf{(E)} ~162\qquad$

(2024 XCMC 10 #23)

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Find the sum of the digits of the unique prime number $p\geq 31$ such that $$\binom{p^2-1}{846}+\binom{p^2-2}{846}$$is divisible by $p$.

$\textbf{(A)} ~7\qquad \textbf{(B)} ~8\qquad \textbf{(C)} ~10\qquad \textbf{(D)} ~11\qquad \textbf{(E)} ~13\qquad$

(2024 XCMC 10 #24)

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$\star$ Alex has a $4$ by $4$ grid of squares. Let $N$ be the number of ways that Alex can fill out each square with one of the letters $A$, $B$, $C$, or $D$ such that in every row and column, the number of $A$'s and $B$'s are the same, and the number of $C$'s and $D$'s are the same. (For example, a row with squares labeled $BDAC$ or $DCCD$ is valid, while a row with squares labeled $ACDA$ or $CBCB$ is not valid.) Find the remainder when $N$ is divided by $7$.

$\textbf{(A)} ~0\qquad \textbf{(B)} ~1\qquad \textbf{(C)} ~3\qquad \textbf{(D)} ~4\qquad \textbf{(E)} ~6\qquad$

(2024 XCMC 10 #25)

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How many ways are there to divide a $4$ by $4$ grid of squares along the gridlines into two or more pieces such that if three pieces meet at a point $P$, then there are actually four pieces with a vertex at $P$? An example is shown below.

IMAGE

(2025 ELMOCOUNTS CDR #19)

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How many ways are there to label each cell of a 4-by-4 grid of squares with either 1, 2, 3, or 4 such that no two adjacent cells have the same label and no two adjacent cells have labels that sum to 5?

(2025 ELMOCOUNTS Sprint #20)

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Let $a,b,c,d,e,f$ be real numbers satisfying the system of equations
$$\begin{cases} a+b+c+d+e+f=1 \\ a+2b+3c+4d+5e+6f=2 \\ a+3b+6c+10d+15e+21f=4 \\ a+4b+10c+20d+35e+56f=8 \\ a+5b+15c+35d+70e+126f=16 \\ a+6b+21c+56d+126e+252f=32. \\ \end{cases}$$What is the value of $a+3b+9c+27d+81e+243f$?

(2025 ELMOCOUNTS Sprint #26)

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There are seven students at a camp. There are seven classes available and each student chooses some of the classes to take. Every student must choose at least two classes. How many ways are there for the students to choose the classes such that each pair of classes has exactly one student in common?

(2025 ELMOCOUNTS Team #8)

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$\star$ In $\triangle{ABC}$, the incircle is tangent to $\overline{BC}$ at $D$, and $E$ is the reflection of $D$ across the midpoint of $\overline{BC}$. Suppose that the inradii of $\triangle ABE$ and $\triangle ACE$ are $4$ and $11$ respectively, and the distance between their incenters is $25$. What is the inradius of $\triangle{ABC}$? Express your answer as a common fraction.

(2025 ELMOCOUNTS Team #10)

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Let $n$ be a positive integer and let $S$ be the set of all $n$-tuples of $0$'s and $1$'s. Two elements of $S$ are said to be neighboring if and only if they differ in only one coordinate. Bob colors the elements of $S$ red and blue such that each blue $n$-tuple is neighboring to exactly two red $n$-tuples and no two red $n$-tuples neighbor each other. If $n>100$, find the least possible value of $n$.

(2025 ELMOCOUNTS Target #6)
22 replies
vincentwant
May 25, 2025
vincentwant
Today at 1:41 AM
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Warning!
VivaanKam   42
N Yesterday at 9:58 PM by Yiyj
This problem will try to trick you! :!:

42 replies
VivaanKam
May 5, 2025
Yiyj
Yesterday at 9:58 PM
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no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5
parmenides51   12
N May 19, 2025 by Yiyj
Find the number of integer solutions to $||x| - 2020| < 5$.
12 replies
parmenides51
Jan 18, 2021
Yiyj
May 19, 2025
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no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5
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parmenides51
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parmenides51
#1 h  Jan 18, 2021, 8:27 AM
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Find the number of integer solutions to $||x| - 2020| < 5$.
This post has been edited 1 time. Last edited by parmenides51, Jan 18, 2021, 8:28 AM
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MrOreoJuice
594 posts
MrOreoJuice
#2 h  Jan 18, 2021, 8:36 AM • 1 Y
Y by Ash_Blanc
sol
This post has been edited 2 times. Last edited by MrOreoJuice, Jan 18, 2021, 8:39 AM
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cubingsoda
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cubingsoda
#3 h  Jan 18, 2021, 8:40 AM • 3 Y
Y by centslordm, a.shyam.25, suvamkonar
enumeration
This post has been edited 2 times. Last edited by cubingsoda, Jan 18, 2021, 9:04 AM
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TuZo
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TuZo
#4 h  Jan 18, 2021, 8:47 AM • 1 Y
Y by centslordm
forester2015 wrote:
sol

Solution posted befor!
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cubingsoda
19220 posts
cubingsoda
#5 h  Jan 18, 2021, 8:51 AM • 3 Y
Y by centslordm, a.shyam.25, suvamkonar
TuZo wrote:
forester2015 wrote:
sol

Solution posted befor!

yes but it is a different soution
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TuZo
19351 posts
TuZo
#6 h  Jan 18, 2021, 9:02 AM • 3 Y
Y by centslordm, Mango247, Mango247
Your "solution" it is not a solution, just an enumeration! ;)
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cubingsoda
19220 posts
cubingsoda
#7 h  Jan 18, 2021, 9:03 AM • 3 Y
Y by centslordm, a.shyam.25, suvamkonar
TuZo wrote:
Your "solution" it is not a solution, just an enumeration! ;)

oh ok i edited
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RamAgarwal2009
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RamAgarwal2009
#8 h  Aug 30, 2023, 1:01 PM
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The answer is 18
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S_14159
48 posts
S_14159
#9 h  Mar 10, 2025, 5:57 PM • 1 Y
Y by Exponent11
We have $-5<|x|-2020<5$, now $|x|>2015$ or $|x|<2025$ which give $\boxed{18}$ number of solutions.
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AshAuktober
1014 posts
AshAuktober
#10 h  Apr 19, 2025, 9:11 AM
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We have $|x| \in \{2016, 2017, \dots, 2024\$, which gives us $\boxed{18}$ solutions. $\square$
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MathematicalArceus
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MathematicalArceus
#11 h  May 19, 2025, 7:03 PM
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Ugh JEE aah problem
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Yiyj
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Yiyj
#12 h  May 19, 2025, 7:18 PM
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Yiyj
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#13 h  May 19, 2025, 7:19 PM
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AshAuktober wrote:
We have $|x| \in \{2016, 2017, \dots, 2024\}$, which gives us $\boxed{18}$ solutions. $\square$
FTFY
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