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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)!

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[*]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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Mathcounts and how to learn

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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
Unique Rational Number Representation
abhisruta03   18
N 5 minutes ago by Reacheddreams
Source: ISI 2021 P3
Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.
18 replies
abhisruta03
Jul 18, 2021
Reacheddreams
5 minutes ago
Math solution
Techno0-8   1
N 24 minutes ago by jasperE3
Solution
1 reply
Techno0-8
3 hours ago
jasperE3
24 minutes ago
D1027 : Super Schoof
Dattier   1
N 32 minutes ago by Dattier
Source: les dattes à Dattier
Let $p>11$ a prime number with $a=\text{card}\{(x,y) \in \mathbb Z/ p \mathbb Z: y^2=x^3+1\}$ and $b=\dfrac 1 {((p-1)/2)! \times ((p-1)/3)! \times ((p-1)/6)!} \mod p$ when $p \mod 3=1$.



Is it true that if $p \mod 3=1$ then $a \in \{b,p-b, \min\{b,p-b\}+p\}$ else $A=p$.
1 reply
Dattier
3 hours ago
Dattier
32 minutes ago
minimizing sum
gggzul   0
34 minutes ago
Let $x, y, z$ be real numbers such that $x^2+y^2+z^2=1$. Find
$$min\{12x-4y-3z\}.$$
0 replies
gggzul
34 minutes ago
0 replies
Geo problem
MathWinner121   10
N 2 hours ago by ethan2011
Using analytical geometry, prove that the sum of the squares of a parallelograms diagonals equal the sum of the squares of the side lengths.
10 replies
MathWinner121
Yesterday at 10:15 PM
ethan2011
2 hours ago
cool math probel i dont know how to solve
Soupboy0   6
N 3 hours ago by iwastedmyusername
Define the derangement sum $D(n)$ to be the sum of all permutations of the digits of $n$ such that no digit appears in its original spot. For example, $D(12) = 21$ and $D(123)=231+312=543$. What is $D(12345)$?
6 replies
Soupboy0
3 hours ago
iwastedmyusername
3 hours ago
Mass points question
Wesoar   2
N 4 hours ago by itsjeyanth
So I was working my way through mass points, and I found a rule that basically says:

"If transversal line EF crosses cevian AD in triangle ABC, you must split mass A into Mass ab and Mass ac. Could someone explain to me why this makes sense/why we couldn't just use mass A?
2 replies
Wesoar
Today at 2:27 AM
itsjeyanth
4 hours ago
Doubt on a math problem
AVY2024   16
N 5 hours ago by maxamc
Solve for x and y given that xy=923, x+y=84
16 replies
AVY2024
Apr 8, 2025
maxamc
5 hours ago
problemo
hashbrown2009   3
N 5 hours ago by maxamc
if x/(3^3+4^3) + y/(3^3+6^3) =1

and

x/(5^3+4^3) + y/(5^3+6^3) =1

find the 2 values of x and y.
3 replies
hashbrown2009
Mar 30, 2025
maxamc
5 hours ago
Geometry Help
ILOVECATS127   0
Today at 1:07 PM
Hello, I needed some help understanding this concept from Chapter 12, Geometry:

Points P, Q and R are on circle O such that

Arc PQ = 78°, arc QR = 123°, and arc PQR = 201°.

1. Find ∠PQO
2. Find ∠POR

Please help me understand HOW to solve these 2 problems.
0 replies
ILOVECATS127
Today at 1:07 PM
0 replies
prime numbers
wpdnjs   111
N Today at 10:54 AM by ostriches88
does anyone know how to quickly identify prime numbers?

thanks.
111 replies
wpdnjs
Oct 2, 2024
ostriches88
Today at 10:54 AM
How many ways to spell the word COUNT
yinglinwu   8
N Today at 3:51 AM by yinglinwu
This is the problem 2.771 from the book Competition Math For Middle School. The printed answer is 16, which is obtained by going through the triangle. I have no idea about what's going on there. Can anyone help explain please?

BTW, my understanding about a way to spell COUNT is simply to choose an order of those letters and write down them one by one at their designated locations, like writing down U first at the 3rd location, then O at the 2nd position, N at 4th, T at 5th, and C at 1st. Therefore, in total we should have 5! ways to spell the word.
8 replies
yinglinwu
May 4, 2025
yinglinwu
Today at 3:51 AM
Easy number theory
britishprobe17   31
N Today at 2:20 AM by martianrunner
The number of factors from 2024 that are greater than $\sqrt{2024}$ are
31 replies
britishprobe17
Oct 16, 2024
martianrunner
Today at 2:20 AM
max number of candies
orangefronted   12
N Today at 1:27 AM by iwastedmyusername
A store sells a strawberry flavoured candy for 1 dollar each. The store offers a promo where every 4 candy wrappers can be exchanged for one candy. If there is no limit to how many times you can exchange candy wrappers for candies, what is the maximum number of candies I can obtain with 100 dollars?
12 replies
orangefronted
Apr 3, 2025
iwastedmyusername
Today at 1:27 AM
integer functional equation
ABCDE   148
N Apr 22, 2025 by Jakjjdm
Source: 2015 IMO Shortlist A2
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$.
148 replies
ABCDE
Jul 7, 2016
Jakjjdm
Apr 22, 2025
integer functional equation
G H J
G H BBookmark kLocked kLocked NReply
Source: 2015 IMO Shortlist A2
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Sedro
5845 posts
#156
Y by
The only solutions are $f(x)=-1$ and $f(x)=x+1$, which clearly work. We now show that there are no other solutions. Let $P(x,y)$ denote the assertion.

Clearly, the only constant solution is $f(x)=-1$, so assume $f$ is nonconstant. By $P(x,f(x))$, we have that $f(x-f(f(x)))=-1$. Let $u=-f(f(0))$; then, $f(u)=-1$. From $P(x,u)$ we obtain $f(x+1)=f(f(x))$. Then, from $P(f(x)-1,x)$, we obtain $f(-1) = f(f(f(x)-1))-f(x)-1$. Since $f(x+1)=f(f(x))$, we have $f(f(f(x)-1)) = f(f(x)) = f(x+1)$. Thus, $f(x+1)-f(x) = f(-1)+1$. The right hand side is constant, so $f$ is linear. Since we assumed $f$ is nonconstant, $f$ is injective, and by $f(x+1)=f(f(x))$, we must have $f(x)=x+1$, as desired. $\blacksquare$
This post has been edited 1 time. Last edited by Sedro, Jul 21, 2024, 10:24 PM
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joshualiu315
2534 posts
#157 • 1 Y
Y by Amir Hossein
The answer is $f(x) = \boxed{-1, x-1}$, which works. Denote the given assertion as $P(x,y)$.

Notice that $P(x,f(x))$ yields

\[f(x-f(f(x))) = -1,\]
so there exists a value $k$ such that $f(k)=-1$. Then, consider $P(x,k)$:

\[f(x+1) = f(f(x)).\]
Plug this back into the original equation and denote the new assertion as $Q(x,y)$. Now, $Q(f(x)+k,x)$ yields

\[f(k)+1 = f(x+k+2)-f(x).\]
Setting $k=-1$ shows that each consecutive difference is constant. Hence, $f$ is linear. Finally, plugging in $f(x) = ax+b$ and solving gives the solution set.
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pie854
243 posts
#158
Y by
Setting $x=y+f(y)$ implies $f(f(y+f(y)))=2f(y)+1$. Setting $y=f(x)$ implies $f(x-f(f(x)))=-1$, setting here $x=y+f(y)$ and simplifying gives $f(f(y-1))=f(y)$, thus $f(f(x))=f(x+1)$ for all $x$. Now set $x=f(y)-1$. Note that $$f(f(f(y)-1))=f(f(y))=f(y+1),$$so we get $f(y+1)=f(y)+(1+f(-1))$. By induction $f(x)=f(0)+x(1+f(-1))$. Plugging back we get the only solutions are $f(x)=x+1$ for all $x$ and $f(x)=-1$ for all $x$.
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RetroFuel
11 posts
#159
Y by
We claim that the only solutions are $f(x) \equiv -1$ and $f(x)=x+1$ for all $x$. Let $P(x,y)$ be the assertion.
$P(x,f(x))$ gives us that $$f(x-f(f(x)))=-1 \forall x$$This implies that either $f(x) \equiv -1$ which we can check to be a working solution or that $x- f(f(x)) = c \forall x$ where c is a constant value and $f(c) = -1$. This means that $f(f(x)) = c+x$.
Subtituting this in our original function we get that, $$f(x-f(y))=x-f(y) + c -1$$$c-1$ is a constant value, and then we can write $x-f(y) = z$. We see that $z$ can range through all integral values as we can fix $y=y_0$ and then we can vary the value of $x$ accordingly to produce all integer values, this means that $$f(x) = x+d \forall x$$where $d$ is a constant value. Substituting this function in our original function gives us that $d=1$. This means that $f(x)=x+1\forall x$ is also a solution which can be checked. Q.E.D
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onyqz
195 posts
#160
Y by
needed a hint for the second claim (I am bad in FE :rotfl: )

solution
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SomeonesPenguin
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#161
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Here is probably the most awkward solution to this problem.

We will show that the only solutions are $f\equiv -1$ and $f(x)=x+1$. From $P(x,f(x))$ we get that there is $a$ such that $f(a)=-1$. $P(x,a)$ gives \[f(f(x))=f(x+1)\]Now substitute this back into the given FE and let $Q(x,y)$ denote the new FE.

If $f$ is injective, we get the solution $f(x)=x+1$. Now if there are $p\neq q$ such that $f(p)=f(q)$, subtracting $P(p,y)$ from $P(q,y)$ yields $f(p-f(y))=f(q-f(y)$. Let $S=\{x|f(y)=x\}$.

Claim: $f\equiv -1$ or there is some positive integer in $S$.
Proof: Suppose that $f\not\equiv -1$, then there are $n>m$ in $S$.

Case 1. $n=m+1$. Take some $x$ and $y$ such that $f(x+1)=n$ and $f(y)=m$. $Q(n,m)$ gives that $0$ is in $S$. If $f(b)=0$, from $Q(x,b)$ we get $f(x)+1=f(x+1)$, hence $1$ is in $S$, which suffices.

Case 2. $n>m+1$. Take $x$ and $y$ such that $f(x+1)=n$ and $f(y)=m$. $Q(x,y)$ gives the desired conclusion. $\square$

Now look back at $f(p-f(y))=f(q-f(y)$. Since there is a positive integer in $S$ and $-1\in S$, it's not hard to see that $f(n)=f(n+c)$ for all $n$, where $c=|p-q|\neq 0$. $Q(y+kc-1,y)$ gives \[-1=f(y+kc-1-f(y)), \ \forall y\]From now, the idea is to find a smaller period of $f$ and induct to get that $f$ has period $1$. Notice that if there are $x$ and $y$ such that $x-f(x)\not\equiv y-f(y)\pmod c$, then by picking suitable $k$s in the above we can arrive at $-1=f(\alpha)=f(\beta)$ with $|\alpha-\beta|\le c-1$ which gives a smaller period.

Now suppose that $x-f(x)$ is constant modulo $c$. Looking at $f(f(x))=f(x+1)$ modulo $c$ gives that $f(x)\equiv x+1\pmod c$. Notice that this implies that $|S|=c$. From $Q(x,y)$ we get that $|S-S|=|S|=c$ (Here $S-S=\{x-y|(x,y)\in S\times S\}$). One can see that $|S-S|$ is at least $2|S|-1$ since we have $|S|-1$ positive and $|S|-1$ negative differences and we have $0$. Hence $c\le 1$, which implies that $f$ is constant. $\blacksquare$
This post has been edited 2 times. Last edited by SomeonesPenguin, Nov 8, 2024, 5:13 PM
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L13832
267 posts
#162 • 1 Y
Y by alexanderhamilton124
Nice one!

Let $P(x,y)$ be the assertion $f(x-f(y))=f(f(x))-f(y)-1$

$P(x,f(x))\implies f(x-f(f(x)))=-1$, so there exists an $a$ such that $f(a)=-1$.

$P(x,a)\implies f(f(f(x)))=f(f(x))\implies f(x+1)=f(f(x))$, if $f$ is injective then $\boxed{f(x)=x+1}$.

$P(x,x)\implies f(x-f(x))=f(x+1)-f(x)-1$, also
$f(x-f(x))= f(f(x-f(x)-1))=f(f(x-f(f(x))))=f(-1)$
so $f(x+1)-f(x)$ is constant giving $f$ to be linear.

If $f$ is constant then $\boxed{f\equiv -1}$ otherwise $f$ is injective. :yoda:
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HamstPan38825
8859 posts
#163
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The answers are $f \equiv -1$ and $f \equiv x+1$, which both work.

Main Proof: Setting $y=f(x)$ yields $f(x-f(f(x)) = -1$, which we call $(1)$. Setting $x = f(y)$ shows $f(0) = f(f(f(y))) - f(y) - 1$, which we call $(2)$. But rearranging terms and applying $(1)$ yields \[-1=f(f(y) - f(f(f(y)))) = f(-1-f(0))\]i.e. for $c = f(0) + 1$, we have $f(-c) = -1$.

Set $y=c$ in the original to get $f(x+1) = f(f(x))$. In other words, $(2)$ rearranges to \[f(y+2) - f(y) = f(0)+1.\]In other words, $f(2k) = ck+f(0)$ and $f(2k+1) = ck+f(1)$ for all integers $k$.

Annoying Cleanup: The rest of the proof is suprisingly annoying. Suppose $c \neq 0$ for now; it follows that there are at most two values $x$ such that $f(x) = -1$. On the other hand, $(1)$ with $x = 2k$ yields $f(2k-ck-f(1)) = -1$. If $c \neq 2$, $2k-ck-f(1)$ can take on infinitely many values, hence we must have $c = 2$. It follows that $f(0) = 1$.

Now let $r = f(1)$. I claim $r = 2$ or $r$ must be odd; this follows because $f(2k+2) = f(2k+r)$ by setting $x = 2k+1$, hence we cannot have $r$ any even number except for $2$. Now assume for the sake of contradiction that $r$ is odd. Setting $x = 2k+1$ and $y = 2\ell + 1$ in the original,
\[2(k-\ell) - r + 2 = f(2k+1-2\ell - r) = 2k+2r-1 - 2\ell - r - 1 = 2(k-\ell) + r-2.\]It follows $r = 2$, contradiction. This case thus yields $f \equiv x+1$.

Now assume $c = 0$. Then $f(0) = -1$, so $f(2k) = -1$ for all $k$. Setting $(x, y) = (0, 1)$ yields $f(-f(1)) = -1$, thus either $f \equiv -1$ on the odds too or $f(1)$ is even. But then by setting $x$ odd and $y = 1$, \[f(1) = f(x-f(1)) = f(f(x)) - f(1) - 1 = -f(1) - 2\]which yields $f(1) = -1$, contradiction. Thus $f \equiv -1$ in this case, and we are finally done.

Remark: I just noticed I solved this problem before in a pretty similar way while also remarking on the annoying nature of the finish. Some things never change.
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Ilikeminecraft
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#164
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The answer is $f\equiv-1, f\equiv x + 1,$ both of which clearly work.
let $P(x, y)$ be the assertion.
$P(x, f(x))\implies f(x - f(f(x))) = f(f(x)) - f(f(x)) -1 = -1.$
$P(x, x - f(f(x)))\implies f(x + 1) = f(x - f(x - f(f(x)))) = f(f(x)) - f(x - f(f(x))) - 1 = f(f(x)).$
$P(x, y), P(x - 1, y)\implies$
\begin{align*}
    f(x - f(y)) & = f(x + 1) - 1 - f(y) \\ 
    f(x - 1 - f(y)) & = f(x) - 1 - f(y) 
\end{align*}By subtracting, we get $f(x - f(y)) - f(x - 1 - f(y)) = f(x + 1) - f(x).$ Taking $x= f(y)$, we get $f(0) - f(-1) = f(f(y) + 1) - f(f(y)) = f(f(f(y))) - f(f(f(y - 1))).$ Hence, $f(f(f(x)))$ is a linear function since its finite difference is constant. We write $f(f(f(x))) = ax + b.$
$P(f(y), y)\implies f(0) = f(f(f(y))) - f(y) - 1\implies f(y) = ay + b + 1 - f(0),$ or $f$ is linear.

let $f(x) = ax + b.$ plug this in to get $ax - a^2y - ab + b = a^2x + ab + b - ay - b - 1.$ Thus, $a = 1, 0.$ If $a = 1, b = 1,$ and if $a = 0, b = -1.$
This post has been edited 1 time. Last edited by Ilikeminecraft, Jan 24, 2025, 11:13 PM
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EpicBird08
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#165
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The only solutions are $f(x) = -1$ and $f(x) = x + 1$, which both work. Now we show that these are all the solutions. Let $P(x,y)$ denote the given assertion.

$P(0,f(0))$ gives $f(-f(f(0))) = -1.$ Letting $u = -f(f(0)),$ we see that $P(x,u)$ gives $$\boxed{f(f(x)) = f(x+1)}.$$Then $P(f(x),x)$ gives $$f(0) = f(f(f(x))) - f(x) - 1 = f(f(x+1)) - f(x) - 1 = f(x+2) - f(x) - 1.$$This implies that $$\boxed{f(x+2) = f(x) + f(0) + 1}.$$We now consider two cases.

Case 1: $f(0) = -1.$ The above boxed equation immediately gives $f(2k) = -1$ for all $k \in \mathbb{Z}.$ Let $f(1) = f(2k+1) = c$ for integers $k.$
Subcase 1: $c$ is odd. Then $P(2k,2k-1)$ gives $c = c - c - 1 \implies c = -1,$ yielding our first solution $f(x) = -1$ for all $x.$
Subcase 2: $c$ is even. Then $P(2k+1,2k+1)$ gives $c = -1 - c - 1 \implies c = -1,$ a contradiction.
Thus this case has been exhausted, and it yields one solution $\boxed{f(x) = -1}.$

Case 2: $f(0) \ne -1.$ Then the first boxed equation implies that if $f(x) \equiv x+1 \pmod{2},$ then $f(x) = x+1.$ By the second boxed equation, we know $f(2) = 2 f(0) + 1$ is odd, as is $2 + 1 = 3.$ Hence $f(2) = 3,$ immediately giving $f(0) = 1.$ Hence $f(x) = x+1$ for even $x.$ Let $f(x) = x + c$ for odd $x.$ Again, the first boxed equation implies $f(1+c) = f(f(1)) = f(2) = 3.$ Hence either $1+c+1 = 3$ or $1 + 2c = 3.$ Both cases give $c = 1,$ yielding our second solution $\boxed{f(x) = x + 1}$ for all $x.$

Hence the only solutions are those claimed at the beginning.
This post has been edited 2 times. Last edited by EpicBird08, Jan 29, 2025, 12:11 AM
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Ilikeminecraft
616 posts
#166
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The solution is $f\equiv 1, x + 1.$

Take $y = f(x)$ to get that there exists $a = x - f(f(x))$ such that $f(a) = -1.$

Take $y = a$ to get that $f(x + 1) = f(f(x)).$

If $f$ is injective, we are done, as this implies $x + 1 = f(x),$ which indeed works.

Now, assume $f$ is not injective, and $f(a) = f(b)$ for some $a\neq b.$ Note that we can get $f(a + 1) = f(f(a)) = f(f(b)) = f(b + 1),$ and so thus, $f(x) = f(x + k(a - b))$ for some $k \in \mathbb Z.$

Take $x = f(y)$ to get $f(0) = f(f(f(y))) - f(y) - 1 = f(y + 2) - f(y) - 1$ to get that $f(2k)$ is linear and $f(2k + 1)$ is linear. By $f(x) = f(x + k(a - b)),$ we can also get that $f(2k) = c_2, f(2k + 1) = c_1$ for two constants(not necessarily the same). Finally, we do parity casework on $c_1, c_2:$
\begin{enumerate}
\item If $c_2\equiv 0\pmod 2,$ then taking $x\equiv y\equiv 0\pmod 2$ tells us $c_2 = -1,$ contradiction.
\item If $c_1 \equiv 0\pmod 2,$ then taking $x\equiv 0, y\equiv 1\pmod 2$ tells us $c_1 = -1,$ contradiction.
\item If $c_1 \equiv c_2 \equiv 1 \pmod 2,$ taking $x\equiv 0\pmod 2$ tells us $f(y) = -1,$ which finishes.
\end{enumerate}
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Bardia7003
20 posts
#167
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Let $P(x,y)$ denote the given assertion.
$\newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}}
P(f(x),x): f(0) = \underline{f(f(f(x))) - f(x) - 1 \quad (\RomanNumeral{1})} \\
P(x, f(x)):  \underline{f(x - f(f(x))) = -1 \quad (\RomanNumeral{2})} \\
P(x, x - f(f(x))): f(x - f(x - f(f(x)))) = f(f(x)) - f(x - f(f(x))) - 1 \xrightarrow{\RomanNumeral{2}}  \underline{f(x+1) = f(f(x)) \quad (\RomanNumeral{3})}$
So by $\newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}} (\RomanNumeral{3})$ we have that $f(f(f(x))) = f(f(x+1)) = f(x+2)$, and putting that in $\newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}} (\RomanNumeral{1})$ we have: $ \underline{f(0) + 1 = f(x+2) - f(x) \newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}} \quad (\RomanNumeral{4})}$. Which means $f(2x)$ is linear. So $f(2x) = ax + b$, which $a = f(0) + 1 = b + 1$, So $f(2x) = (b+1)x + b$.
$f(2x+1)$ is also linear, so $f(2x + 1) = ax + c (a = f(0) + 1)$
We want to prove that $b$ is odd, so we assume otherwise. For an odd $x$, $f(f(2x)) = f((b+1)x + b)$, $(b+1)x$ is odd and $b$ is even so the value is odd: $f(f(2x)) = (b+1)(\frac{(b+1)x + b - 1}{2}) + c$. Also by $\newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}} (\RomanNumeral{3})$ we know $f(f(2x)) = f(2x + 1) = (b+1)x + c$
So $(b+1)x = (b+1)(\frac{(b+1)x + b - 1}{2})$. $b$ is even so $b+1\neq0$ and we can cross it out: $x = \frac{bx + x + b - 1}{2} \rightarrow x = bx + b -1 \rightarrow (1-b)x = (b - 1) \xrightarrow{b-1\neq0} x = -1$, but we had the equation for any odd $x$, hence contradiction. As a result, we proved $b$ is odd.
Now we know $f(2x+1) = ax+c \xrightarrow{\newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}} \RomanNumeral{3}} f(f(2x)) = ax + c \rightarrow f(ax + b) = ax + c \xrightarrow{x := 0} f(b) = c$
As we know $b$ is odd, so:
$c = f(b) = a(\frac{b-1}{2}) + c \rightarrow a(\frac{b-1}{2}) = 0$ so $b = 1$ or $a = 0$.
Case 1: $a = 0$. Then $b+1 = 0, f(2x) = b \rightarrow f(2x) = -1$ and $f(2x + 1) = c$. If c is odd, then by $\newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}} (\RomanNumeral{3})$ we know $f(c+1) = f(f(c)) = f(c) = c$ and $c + 1$ is even so $-1 = c = f(2x+1)$.
Hence $\boxed{f(x) = -1 \quad \forall x \in \mathbb{Z}}$ is the solution in this case, which indeed works.
Case 2: $b=1$. Then $f(2x) = 2x + 1, f(2x + 1) = 2x + c$. By $\newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}} (\RomanNumeral{2})$ we know $\newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}} f(2x - f(f(2x)) = -1 \xrightarrow{\RomanNumeral{3}} f(2x - f(2x + 1)) = -1 \rightarrow f(2x - 2x - c) = -1 \rightarrow f(-c) = -1$. If c is even then $-c +1 = -1 \rightarrow c = 2$, and we want to prove $c$ is even so we can conclude $c=2$, so we assume not. If c is odd then $f(f(c)) = f((c-1) + c) = ((2c-1)-1) + c$, and by $\newcommand{\RomanNumeral}[1]{\MakeUppercase{\romannumeral #1}} (\RomanNumeral{3})$ we know that $f(f(c)) = f(c+1) = (c+1) + 1$ so $3c - 2 = c + 2 \rightarrow c = 2$ but we supposed c is odd, contradiction.
So we proved $c$ is even and as a result $c=2$, as we proved. Now we have that $f(2x) = 2x+1, f(2x+1) = 2x+2$, so we can generally conclude $\boxed{f(x) = x+1 \quad \forall x \in \mathbb{Z}}$ in this case, which is also a solution.
So both cases are solved and we found all the solutions. $\blacksquare$
Please feel free to point out if I've made any mistakes through this proof, thanks. :)
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Ihatecombin
60 posts
#168
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A pretty unique problem, bit annoying to be honest lol. The only solutions are $f(x) \equiv -1$ and $f(x) \equiv x+1$.
You can use \(P(x,f(x))\) to get
\[f(x - f^2(x)) = -1\]Afterwards let \(f(r) = -1\), we can use \(P(x,r)\)
to get
\[f(x+1) = f^2(x)\]Notice that the equation can be transformed into
\[f(x-f(y)) = f(x+1) - f(y) - 1\]Let \(f(p+1) = k \neq -1\) (otherwise \(f(x) \equiv -1\), which works).
Notice that if we use \(P(k+p+1,p+1)\), then we must have
\[f(p+1) = f(k+p+2) -k-1 \Longrightarrow 2k+1 =  f(k+p+2)\]Since \(k \neq -1\), we must have that \(2k+1 \neq k\) and \(2k+1\) is also in the image of \(f(x)\),
thus \(2(2k+1) + 1\) is also in the image and so on. Therefore the set containing all the integers which are in the image of \(f(x)\) is infinite.
Now let \(n\) be a number such that \(f(n) = k \neq -1\) for some \(k\). Substituting \(P(x,n)\) we have
\[f(x-k) = f(x+1)-k-1\]we already know that if \(k\) is in the image of \(f(x)\), then \(2k+1\) is in the image. Let \(f(m) = 2k+1\), we can substitute \(P(x,m)\) to get
\[f(x-2k-1) = f(x+1) - 2k - 2\]Thus we must have
\[f(x-2k-1) + k + 1= f(x-k) \Longrightarrow f(x) = f(x-k-1) +k+1\]for some \(k \neq -1\).

Notice that since \(f(x+1) = f^2(x)\), if \(f(x)\) is injective then we are done \(f(x) \equiv x+1\).
Therefore assume \(f(x) \not\equiv -1\) and \(f(x) \not\equiv x+1\) and \(f(x)\) is not injective. Let \(f(a) = f(b)\), notice that since \(f(x+1) = f^2(x)\), we must have
\[f(a+1) = f^2(a) = f^2(b) = f(b+1)\]similarly
\[f(a+2) = f^2(a+1) = f^2(b+1) = f(b+2) \Longrightarrow f(a) = f(b) = f(2b-a)\]By induction we have
\[f(a) = f(xb-(x-1)a)\]However substituting \(k+1\), we have
\[f(a) = f([k+1](b-a) + a)\]But since \(f(x) = f(x-k-1) + k +1\), we easily get
\[f(a) = f([k+1](b-a) + a) = f(a) + (k+1)(b-a)\]This is a clear contradiction.
This post has been edited 1 time. Last edited by Ihatecombin, Mar 13, 2025, 1:28 PM
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Adywastaken
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#169
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$y=f(x)\implies f(x-f(f(x)))=-1$
$y=x-f(f(x))\implies f(x+1)=f(f(x))$
$x=f(y)-1\implies f(-1)+1=f(f(f(x)-1))-f(x)=f(f(x))-f(x)=f(x+1)-f(x)$
So, $f=mx+c$, and matching coefficients,
$m^2=m, 2mc=c+1$
So, $(m,c)=(0,-1),(1,0)$
$f(x) \equiv -1$ or $f(x)=x+1$
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Jakjjdm
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#170
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The only solutions are $f(x) \equiv -1$ and $f(x) = x + 1$. Let $P(m,n)$ be the main equation plugging $x = m$ and $y = n$. $P(x,f(x)) : f(x - f(f(x))) = - 1$, so let k be an integer such that $f(k) = -1$, so $P(x,k) : f(x +1) = f(f(x))$. $P(f(y) - 1, y) : f(-1) + 1 = f(f(f(y) - 1)) - f(y) = f(y + 1) - f(y)$, so the function is linear. Now, Just check the cases when $f(x) \equiv c$, giving $f(x) \equiv -1$, and check when $f(x) = x + c$, that gives $f(x) = x + 1$, and we're done.
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