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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
6 hours ago
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
6 hours ago
0 replies
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k i A Letter to MSM
Arr0w   23
N Sep 19, 2022 by scannose
Greetings.

I have seen many posts talking about commonly asked questions, such as finding the value of $0^0$, $\frac{1}{0}$,$\frac{0}{0}$, $\frac{\infty}{\infty}$, why $0.999...=1$ or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
[list]
[*]Firstly, the case of $0^0$. It is usually regarded that $0^0=1$, not because this works numerically but because it is convenient to define it this way. You will see the convenience of defining other undefined things later on in this post.

[*]What about $\frac{\infty}{\infty}$? The issue here is that $\infty$ isn't even rigorously defined in this expression. What exactly do we mean by $\infty$? Unless the example in question is put in context in a formal manner, then we say that $\frac{\infty}{\infty}$ is meaningless.

[*]What about $\frac{1}{0}$? Suppose that $x=\frac{1}{0}$. Then we would have $x\cdot 0=0=1$, absurd. A more rigorous treatment of the idea is that $\lim_{x\to0}\frac{1}{x}$ does not exist in the first place, although you will see why in a calculus course. So the point is that $\frac{1}{0}$ is undefined.

[*]What about if $0.99999...=1$? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
\begin{align*} \sum_{n=1}^{\infty} \frac{9}{10^n}&=9\sum_{n=1}^{\infty}\frac{1}{10^n}=9\sum_{n=1}^{\infty}\biggr(\frac{1}{10}\biggr)^n=9\biggr(\frac{\frac{1}{10}}{1-\frac{1}{10}}\biggr)=9\biggr(\frac{\frac{1}{10}}{\frac{9}{10}}\biggr)=9\biggr(\frac{1}{9}\biggr)=\boxed{1} \end{align*}
[*]What about $\frac{0}{0}$? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[/list]
Hopefully all of these issues and their corollaries are finally put to rest. Cheers.

2nd EDIT (6/14/22): Since I originally posted this, it has since blown up so I will try to add additional information per the request of users in the thread below.

INDETERMINATE VS UNDEFINED

What makes something indeterminate? As you can see above, there are many things that are indeterminate. While definitions might vary slightly, it is the consensus that the following definition holds: A mathematical expression is be said to be indeterminate if it is not definitively or precisely determined. So how does this make, say, something like $0/0$ indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that $0/0$ is an indeterminate form.

But we need to more specific, this is still ambiguous. An indeterminate form is a mathematical expression involving at most two of $0$, $1$ or $\infty$, obtained by applying the algebraic limit theorem (a theorem in analysis, look this up for details) in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being calculated. This is why it is called indeterminate. Some examples of indeterminate forms are
\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function $f:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}$ given by the mapping $x\mapsto \sqrt{x}$ is undefined for $x<0$. On the other hand, $1/0$ is undefined because dividing by $0$ is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context.

WHEN THE WATERS GET MUDDIED

So with this notion of indeterminate and undefined, things get convoluted. First of all, just because something is indeterminate does not mean it is not undefined. For example $0/0$ is considered both indeterminate and undefined (but in the context of a limit then it is considered in indeterminate form). Additionally, this notion of something being undefined also means that we can define it in some way. To rephrase, this means that technically, we can make something that is undefined to something that is defined as long as we define it. I'll show you what I mean.

One example of making something undefined into something defined is the extended real number line, which we define as
\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let $-\infty\le a\le \infty$ for each $a\in\overline{\mathbb{R}}$ which means that via this order topology each subset has an infimum and supremum and $\overline{\mathbb{R}}$ is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In $\overline{\mathbb{R}}$ it is perfectly OK to say that,
\begin{align*} a + \infty = \infty + a & = \infty, & a & \neq -\infty \\ a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\ \frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\ \frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\ \frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0). \end{align*}So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,
\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define $\infty \times 0=0$ as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by $0$ is undefined still! Is there a place where it isn't? Kind of. To do this, we can extend the complex numbers! More formally, we can define this extension as
\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, $\tilde{\infty}$ means complex infinity, since we are in the complex plane now. Here's the catch: division by $0$ is allowed here! In fact, we have
\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]where $\tilde{\infty}/\tilde{\infty}$ and $0/0$ are left undefined. We also have
\begin{align*} z+\tilde{\infty}=\tilde{\infty}, \forall z\ne -\infty\\ z\times \tilde{\infty}=\tilde{\infty}, \forall z\ne 0 \end{align*}Furthermore, we actually have some nice properties with multiplication that we didn't have before. In $\mathbb{C}^*$ it holds that
\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]but $\tilde{\infty}-\tilde{\infty}$ and $0\times \tilde{\infty}$ are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with $\mathbb{C}^*$, by defining them as
\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]They behave in a similar way to the Riemann Sphere, with division by $0$ also being allowed with the same indeterminate forms (in addition to some other ones).
23 replies
Arr0w
Feb 11, 2022
scannose
Sep 19, 2022
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k i Marathon Threads
LauraZed   0
Jul 2, 2019
Due to excessive spam and inappropriate posts, we have locked the Prealgebra and Beginning Algebra threads.

We will either unlock these threads once we've cleaned them up or start new ones, but for now, do not start new marathon threads for these subjects. Any new marathon threads started while this announcement is up will be immediately deleted.
0 replies
LauraZed
Jul 2, 2019
0 replies
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k i Basic Forum Rules and Info (Read before posting)
jellymoop   368
N May 16, 2018 by harry1234
f (Reminder: Do not post Alcumus or class homework questions on this forum. Instructions below.) f
Welcome to the Middle School Math Forum! Please take a moment to familiarize yourself with the rules.

Overview:
[list]
[*] When you're posting a new topic with a math problem, give the topic a detailed title that includes the subject of the problem (not just "easy problem" or "nice problem")
[*] Stay on topic and be courteous.
[*] Hide solutions!
[*] If you see an inappropriate post in this forum, simply report the post and a moderator will deal with it. Don't make your own post telling people they're not following the rules - that usually just makes the issue worse.
[*] When you post a question that you need help solving, post what you've attempted so far and not just the question. We are here to learn from each other, not to do your homework. :P
[*] Avoid making posts just to thank someone - you can use the upvote function instead
[*] Don't make a new reply just to repeat yourself or comment on the quality of others' posts; instead, post when you have a new insight or question. You can also edit your post if it's the most recent and you want to add more information.
[*] Avoid bumping old posts.
[*] Use GameBot to post alcumus questions.
[*] If you need general MATHCOUNTS/math competition advice, check out the threads below.
[*] Don't post other users' real names.
[*] Advertisements are not allowed. You can advertise your forum on your profile with a link, on your blog, and on user-created forums that permit forum advertisements.
[/list]

Here are links to more detailed versions of the rules. These are from the older forums, so you can overlook "Classroom math/Competition math only" instructions.
Posting Guidelines
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What belongs on this forum?
How do I write a thorough solution?
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Mathcounts and how to learn

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

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

You can also view marathons via the Marathon tag.

Think this list is incomplete or needs changes? Let the mods know and we'll take a look.
368 replies
jellymoop
May 8, 2015
harry1234
May 16, 2018
J
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IMO Shortlist 2014 N4
hajimbrak   74
N 9 minutes ago by LenaEnjoyer
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)

Proposed by Hong Kong
74 replies
hajimbrak
Jul 11, 2015
LenaEnjoyer
9 minutes ago
J
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USAMO 2003 Problem 1
MithsApprentice   71
N an hour ago by cubres
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
71 replies
MithsApprentice
Sep 27, 2005
cubres
an hour ago
J
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IMO Genre Predictions
ohiorizzler1434   76
N an hour ago by aidan0626
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
76 replies
ohiorizzler1434
May 3, 2025
aidan0626
an hour ago
J
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IMO ShortList 2008, Number Theory problem 2
April   40
N 2 hours ago by ezpotd
Source: IMO ShortList 2008, Number Theory problem 2, German TST 2, P2, 2009
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i + a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.

Proposed by Mohsen Jamaali, Iran
40 replies
April
Jul 9, 2009
ezpotd
2 hours ago
J
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MIT PRIMES STEP
pingpongmerrily   6
N 2 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
2 hours ago
J
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Trivial Factoring MathCounts Question? 2025 MathCounts National Sprint Round #29
ilikemath247365   40
N 4 hours ago by deeptisidana
What is the value of the expression below?

$\frac{(1! + 2! + 3!)(2! + 3! + 4!)(3! + 4! + 5!)...(98! + 99! + 100!)}{(1! - 3(2!) + 3!)(2! - 3(3!) + 4!)(3! - 3(4!) + 5!)...(98! - 3(99!) + 100!)}$.
40 replies
ilikemath247365
May 28, 2025
deeptisidana
4 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
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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.

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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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Math Wizard 2024 Orals S4
PikaVee   1
N Yesterday at 9:55 PM by ZMB038
Define the binary operations a diamond b = ((a + b) mod a) / (ab mod 7) and a heart b = (a − b − 3) / (ab mod 7). What is the value of (50 diamond 2024) heart 50?


”Solution”
1 reply
PikaVee
Yesterday at 2:26 PM
ZMB038
Yesterday at 9:55 PM
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Hard Functional Equation
yaybanana   3
N Apr 10, 2025 by yaybanana
Source: own
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ , s.t :

$f(y^2+x)+f(x+yf(x))=f(y)f(y+x)+f(2x)$

for all $x,y \in \mathbb{R}$
3 replies
yaybanana
Apr 9, 2025
yaybanana
Apr 10, 2025
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Hard Functional Equation
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Reveal topic tags
functional equation
function
algebra
Functional equation in R
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Source: own
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yaybanana
10 posts
yaybanana
#1 h  Apr 9, 2025, 3:35 PM • 2 Y
Y by PikaPika999, Rayanelba
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ , s.t :

$f(y^2+x)+f(x+yf(x))=f(y)f(y+x)+f(2x)$

for all $x,y \in \mathbb{R}$
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Rayanelba
20 posts
Rayanelba
#2 h  Apr 9, 2025, 8:31 PM
Y by
Cool F.E , source?
This post has been edited 1 time. Last edited by Rayanelba, Apr 12, 2025, 9:44 PM
Reason: .
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internationalnick123456
135 posts
internationalnick123456
#3 h  Apr 10, 2025, 3:59 AM • 1 Y
Y by Rayanelba
If $f$ is constant, then $f\equiv 0$ or $f\equiv 1$.
Consider a non-constant function $f$ that satisfies the functional equation.
Let $P(x,y)$ denote the assertion of the functional equation.
Claim 1. $f(0) = 0$Proof
Claim 2. $f(2x) = 2f(x),\forall x\in\mathbb R$ and $f(x^2) = f^2(x),\forall x\in\mathbb R$ Proof
Claim 3. $f(x) \neq 0,\forall x\neq 0$. Proof
Claim 4. $f(1) = 1$ and $f(-1) = -1$ Proof
Claim 5. $f(n) = n,\forall n\in\mathbb N$ Proof
Claim 6. $f(x+3) + f(x-3) = 2f(x),\forall x\in\mathbb R$ Proof
Claim 7. $f(x + 1) + f(x - 1) = 2f(x),\forall x\in\mathbb R$. Proof
Claim 8.$f(x) = x,\forall x\in\mathbb R$Proof
This post has been edited 2 times. Last edited by internationalnick123456, Apr 10, 2025, 4:02 AM
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yaybanana
10 posts
yaybanana
#4 h  Apr 10, 2025, 4:04 AM • 1 Y
Y by internationalnick123456
lmao you actually proved it the exact same way as me
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