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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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0 replies
jlacosta
Mar 2, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Divisibility of a triple
goodar2006   52
N 10 minutes ago by zuat.e
Source: Iran TST 2013-First exam-2nd day-P5
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?

Proposed by Mahan Malihi
52 replies
goodar2006
Apr 19, 2013
zuat.e
10 minutes ago
Rectangle EFGH in incircle, prove that QIM = 90
v_Enhance   61
N 39 minutes ago by daixiahu
Source: Taiwan 2014 TST1, Problem 3
Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.
61 replies
v_Enhance
Jul 18, 2014
daixiahu
39 minutes ago
USAMO PYQ
JetFire008   1
N an hour ago by v_Enhance
Source: USAMO 1993
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals intersecting at $O$. Prove that the reflections of $O$ across $AB$, BC, CD, DA$ are concyclic.
1 reply
JetFire008
3 hours ago
v_Enhance
an hour ago
ABC is similar to XYZ
Amir Hossein   50
N an hour ago by daixiahu
Source: China TST 2011 - Quiz 2 - D2 - P1
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.
50 replies
Amir Hossein
May 20, 2011
daixiahu
an hour ago
{x_i-a}+{x_{i+1}-b} at most 1/2024
navi_09220114   1
N an hour ago by YaoAOPS
Source: Malaysian SST 2023 P5
Find the maximal value of $c>0$ such that for any $n\ge 1$, and for any $n$ real numbers $x_1, \cdots, x_n$ there exists real numbers $a ,b$ such that $$\{x_i-a\}+\{x_{i+1}-b\}\le \frac{1}{2024}$$for at least $cn$ indices $i$. Here, $x_{n+1}=x_1$ and $\{x\}$ denotes the fractional part of $x$.

Proposed by Wong Jer Ren
1 reply
navi_09220114
Aug 27, 2023
YaoAOPS
an hour ago
Nice, simple geo
MathSaiyan   1
N an hour ago by Double07
Source: PErA 2025/4
Let \( ABC \) be an acute-angled scalene triangle. Let \( B_1 \) and \( B_2 \) be points on the rays \( BC \) and \( BA \), respectively, such that \( BB_1 = BB_2 = AC \). Similarly, let \( C_1 \) and \( C_2 \) be points on the rays \( CB \) and \( CA \), respectively, such that \( CC_1 = CC_2 = AB \). Prove that if \( B_1B_2 \) and \( C_1C_2 \) intersect at \( K \), then \( AK \) is parallel to \( BC \).
1 reply
MathSaiyan
5 hours ago
Double07
an hour ago
A geometry problem from the TOT
Invert_DOG_about_centre_O   8
N an hour ago by daixiahu
Source: Tournament of towns Spring 2018 A-level P4
Let O be the center of the circumscribed circle of the triangle ABC. Let AH be the altitude in this triangle, and let P be the base of the perpendicular drawn from point A to the line CO. Prove that the line HP passes through the midpoint of the side AB. (6 points)

Egor Bakaev
8 replies
Invert_DOG_about_centre_O
Mar 10, 2020
daixiahu
an hour ago
IMO Shortlist 2013, Geometry #2
lyukhson   76
N an hour ago by imagien_bad
Source: IMO Shortlist 2013, Geometry #2
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
76 replies
lyukhson
Jul 9, 2014
imagien_bad
an hour ago
IMO Shortlist 2013, Number Theory #3
lyukhson   46
N an hour ago by hgomamogh
Source: IMO Shortlist 2013, Number Theory #3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
46 replies
lyukhson
Jul 10, 2014
hgomamogh
an hour ago
Functional equation in three variables
mavropnevma   17
N 2 hours ago by Ilikeminecraft
Source: Balkan MO 2013, Problem 3
Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied:

(a) $xf(x,y,z) = zf(z,y,x)$,

(b) $f(x, ky, k^2z) = kf(x,y,z)$,

(c) $f(1, k, k+1) = k+1$.

(United Kingdom)
17 replies
mavropnevma
Jun 30, 2013
Ilikeminecraft
2 hours ago
diophantine
hxtung   36
N 2 hours ago by shendrew7
Source: IMO 1997, Problem 5, IMO Shortlist 1997, Q17
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} = b^a$.
36 replies
hxtung
Oct 13, 2003
shendrew7
2 hours ago
USAMO 2002 Problem 4
MithsApprentice   88
N 2 hours ago by Maximilian113
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 - y^2) = x f(x) - y f(y)  \] for all pairs of real numbers $x$ and $y$.
88 replies
MithsApprentice
Sep 30, 2005
Maximilian113
2 hours ago
Functional Equations Marathon March 2025
Levieee   24
N 2 hours ago by maromex
1. before posting another problem please try your best to provide the solution to the previous solution because we don't want a backlog of many problems
2.one is welcome to send functional equations involving calculus (mainly basic real analysis type of proofs) as long it is of the form $\text{"find all functions:"}$
24 replies
Levieee
Today at 1:03 AM
maromex
2 hours ago
How many digits across all multiples of 1829?
MarkBcc168   13
N 2 hours ago by OronSH
Source: ISL 2022 N5
For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$.

Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.
13 replies
MarkBcc168
Jul 9, 2023
OronSH
2 hours ago
Problem 4
teps   73
N Today at 6:34 AM by Nari_Tom
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here $\mathbb{Z}$ denotes the set of integers.)

Proposed by Liam Baker, South Africa
73 replies
teps
Jul 11, 2012
Nari_Tom
Today at 6:34 AM
Problem 4
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G H BBookmark kLocked kLocked NReply
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bobthegod78
2982 posts
#69
Y by
Why is the solution set so weird.

Here's a walkthrough:
1. P(0,0,0) implies f(0)=f(0)
2. P(a, -a, 0) implies f is even
3. We can rewrite the condition to work for all $a+b=c$.
4. P(1,1,2) gives $4f(1)=f(2)$.
5. If $f(2)=0$, a straightforward induction shows that $f(2x)=0$ for all integers $x$ and $f(2x-1)=c$ for some constant $c$.
6. If $f(2)=4f(1)$, then $P(1,2,3)$ gives us $f(3)^2-10af(3) + 9a^2 =0$ or $(f(3)-9a)(f(3)-a)=0$, implying that $f(3)=a,9a$. If $f(3)=a$, we can induct to get $f(x) =0$ for $x\equiv 0 \pmod 4$, $f(x)=c$ for $x\equiv 1\pmod 2$, and $f(x)=4c$ for $x\equiv 2 \pmod 4$. If $f(3)=9a$, then we can induct to get $f(x) = ax^2$, so another solution is $f(x)=cx^2$.
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cj13609517288
1865 posts
#70
Y by
Setup, using homogeneous

Defining and giving value to g(x) substitution

Casework

Finishing
This post has been edited 8 times. Last edited by cj13609517288, Dec 1, 2022, 4:45 PM
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Taco12
1757 posts
#71
Y by
writing this up from before because I'm bored
We claim the only solutions are

$$f(x)=\begin{cases}0\qquad x \equiv 0 \pmod 2 \\ c\qquad x \equiv 1 \pmod 2\end{cases}$$$$f(x)=\begin{cases}0\qquad x\equiv 0\pmod 4 \\ c\qquad x\equiv 1,3\pmod 4 \\ 4c\qquad x\equiv 2\pmod 4\end{cases}$$$$f(x)=cx^2.$$
Now, we show they are the only ones. Let $P(a,b,c)$ be the assertion. Then,

$P(0,0,0) \rightarrow f(0)=0$.
$P(a,-a, 0) \rightarrow f(a)=f(-a) \rightarrow f \text{ even }$.
$P(a,a,-2a) \rightarrow f(2a) = 0 \text{ or } f(2a) = 4f(a)$.

Claim: If $f(x)=0$, then $f(cx) = 0$ for all integers $c$.
Proof. Use induction, with the base case trivial. Assume $f(x)=f(2x)=\dots=f(cx-c)=0$. Then, $$P(x, cx-c, -cx) \rightarrow f(cx)=0. \square$$
Now we casework on $f(1), f(2), f(4)$.

Case 1: $f(2)=0, f(1) \neq 0$.
By the claim, $f(x)=0$ for even $x$. Let $n,m$ be odd integers. Then $$P(n,m,1-n-m) \rightarrow f(n)=f(m),$$which means $f(x)=c$ for odd $x$.

Case 2: $f(4)=0, f(1), f(2) \neq 0$.
We have $f(x)=0$ for $x \equiv 0 \pmod 4$. Let $n \equiv 1 \pmod 4, m \equiv 3 \pmod 4$. Then $$P(n,m,-n-m) \rightarrow f(n)=f(m),$$which implies $f(x)$ is constant for odd $x$. Now, let $n\equiv 2 \pmod 4$. Note that $f(n) \neq 0$, as otherwise $P(-2, 2-n, n)$ would yield a contradiction. Thus, $f(x)=4x$ for $x \equiv 2 \pmod 4$.

Case 3: $f(1), f(2), f(4) \neq 0$.
Note that $f(2) = 4f(1), f(4) = 16f(1)$. Then, $$P(-1, -2, 3) \rightarrow f(3) = f(1) \text{ or } f(3) = 9f(1).$$Similarly, $$P(-1,-3,4) \rightarrow f(3) = 9f(1) \text{ or } f(3) = 25f(1),$$and combining gives $f(3)=9f(1)$. We can use a similar argument inductively to show that $f(x)=cx^2$.

Case 4: $f(1)=0$.
This implies $f(x)=0$ which falls under the solution previously found in Case 3.

Since all the solutions work and we have exhausted all cases, we are done. $\blacksquare$
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fclvbfm934
759 posts
#72
Y by
Let $P(a, b, c)$ denote the assertion. Quick facts:
\begin{align*}
P(0, 0, 0): f(0) &= 0 \\
P(a, -a, 0): f(a) &= f(-a)
\end{align*}so we get that $f$ is even. Letting $c = - a - b$, we have
$$f(a)^2 + f(b)^2 + f(a + b)^2 = 2f(a)f(b) + 2f(a+b)[f(a) + f(b)].$$Using the quadratic formula to solve for $f(a + b)$ gives us
$$f(a + b) = f(a) + f(b) \pm 2 \sqrt{f(a)f(b)} \qquad{(1)}.$$This tells us a couple of things. The $\sqrt{f(a)f(b)}$ tells us that $f$ must have the same sign everywhere. Furthermore, if we allow $b = 1$, then this means
$$f(a) = \frac{g(a)^2}{f(1)}$$for some $g(a) \ge 0$, unless $f(1) = 0$. Note that if $f(1) = 0$, then by Equation (1), we have $f(a + 1) = f(a)$, so $f(a) = 0$ everywhere. So assume that $f(1) \neq 0$.

Replacing all $f$s with $g$s in Equation (1) yields
\begin{align*}
g(a + b)^2 &= g(a)^2 + g(b)^2 \pm 2g(a)g(b) \\
\Longrightarrow g(a + b) &= |g(a) \pm g(b)| \qquad{(2)}.
\end{align*}We know $g(0) = 0$, and let $g(1) = k$ for some positive integer $k$. Let $Q(a, b)$ denote plugging in $a$ and $b$ into equation (2).

By $Q(1, 1)$, we have $g(2) = |k \pm k|$, so we have two cases.

Case 1: $g(2) = 0$. Using $Q(x, 2)$ gives us $g(x + 2) = g(x)$, and so we have $g(x) = 0$ for even $x$ and $g(x) = k$ for odd $x$.

Case 2: $g(2) = 2k$. By $Q(2, 1)$, we have $g(3) = k$ or $g(3) = 3k$.

Case 2.1: $g(3) = k$. Then, $Q(3, 1)$ tells us $g(4) = 0$ or $2k$. But if $g(4) = 2k$, then $Q(2, 2)$ is a contradiction. So, $g(4) = 0$, and now by $Q(x, 4)$, we get that $g$ is periodic with period $4$, and this determines all of $g$.

Case 2.2: $g(3) = 3k$. We shall use induction to show that $g(n) = nk$ for all positive $n$, with the base case of the first three $n$ handled. Suppose $g(n - 2) = (n-2)k$
and $g(n - 1) = (n-1)k$. Then we have
\begin{align*}
Q(n - 2, 2): g(n) &= nk \text{  or  } (n - 3)k \\
Q(n - 1, 1): g(n) &= nk \text{  or  } (n - 2)k
\end{align*}Notice that there are no absolute value signs here because $n \ge 3$. The only overlapping value for $g(n)$ is $nk$, and so we complete our induction.

Solutions: Wrapping up, we have shown that the only admissible solutions are
  • $g(x) = 0$
  • $g(x) = k$ for $x \equiv 1 \pmod{2}$ and $0$ otherwise
  • $g(0) = 0$, $g(1) = k$, $g(2) = 2k$, and $g(3) = k$, and $g(x) = g(x \pmod{4})$ for all other $x$.
  • $g(x) = kx$ for all $x$.
Then, take $f(x) = c \cdot g(x)$ for any integer $c$. It is easy to see that all of these are valid solutions.
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BlazingMuddy
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There are three classes of solutions to this functional equations; one of them is the nice polynomial function $x \mapsto cx^2$. The other two classes of solutions can be viewed as a function induced from a solution to the same functional equation but with signatures $\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}$, respectively. In general, there is one missing class, induced from a solution with signature $(\mathbb{Z}/2\mathbb{Z})^2 \to \mathbb{Z}$. Of course, it doesn't appear in this problem since $(\mathbb{Z}/2\mathbb{Z})^2$ is not a group quotient of $\mathbb{Z}$.
Quote:
Let $(G, +)$ be an abelian group and $R$ be an integral domain (with $char(R) \neq 2, 3$). Say that $(a, b, c) \in R^3$ is a Heron triple if $a^2 + b^2 + c^2 = 2ab + 2bc + 2ca$. Find all functions $f : G \to R$ such that $(f(x), f(y), f(z))$ is a Heron triple for any $x, y, z \in G$ with $x + y + z = 0$.
The case $char(R) = 2$ is not very interesting on itself; the answers are precisely the group homomorphisms.
On the other hand, the case $char(R) = 3$ to be more complicated than the case $char(R) \neq 3$, since we don't even have $f(0) = 0$ guaranteed anymore!

Answer. A function $f : G \to R$ satisfies the desired properties iff it has one of the following properties:
  • there exists $c \in R$ non-zero and a group homomorphism $\phi : G \to R$ such that $c f(x) = \phi(x)^2$ for any $x \in G$;
  • there exists $c \in R$ and a surjective group homomorphism $\phi : G \to \mathbb{Z}/2\mathbb{Z}$ such that
    \[ f(x) = \begin{cases} 0, & \phi(x) = 0, \\ c, & \phi(x) = 1; \end{cases} \]
  • there exists $c \in R$ and a surjective group homomorphism $\phi : G \to \mathbb{Z}/4\mathbb{Z}$ such that
    \[ f(x) = \begin{cases} 0, & \phi(x) = 0, \\ c, & \phi(x) = \pm 1, \\ 4c, & \phi(x) = 2; \end{cases} \]
  • there exists a Heron triple $(a, b, c) \in R^3$ and a surjective group homomorphism $\phi : G \to (\mathbb{Z}/2\mathbb{Z})^2$ such that
    \[ f(x) = \begin{cases} 0, & \phi(x) = (0, 0), \\ a, & \phi(x) = (1, 0), \\ b, & \phi(x) = (0, 1), \\ c, & \phi(x) = (1, 1). \end{cases} \]
    Since $R$ is an integral domain with $char(R) \neq 3$, for any $r \in R$, $(r, r, r)$ is a Heron triple iff $3r^2 = 0 \iff r = 0$. In particular, $f(0) = 0$.
    Since $R$ is an integral domain, for any $r, s \in R$, $(0, r, s)$ is a Heron triple iff $r = s$. Together with $f(0) = 0$, plugging $x = 0$ and $z = -y$ gives $f(y) = f(-y)$ for any $y \in G$; i.e., $f$ is even. Note that this only requires $f(0) = 0$, but not $char(R) \neq 3$. In particular, the original condition is now equivalent to $(f(x), f(y), f(x + y))$ being a Heron triple for any $x, y \in G$.
    Some parts of the solution will assume that $char(R) \neq 3$, and some parts only assume that $f(0) = 0$.

    First, notice that if $f(x) = 0$, then $(f(x + y), f(y), 0)$ is a Heron triple and thus $f(x + y) = f(y)$ for any $y \in G$. This means that $f^{-1}(0)$ is a subgroup of $G$, and that we have a well-defined induced map $\tilde{f} : G/f^{-1}(0) \to R$ satisfying $\tilde{f}([x]) = f(x)$ for any $x \in G$, and one can check that $\tilde{f}$ is good as well, i.e., $(\tilde{f}(x), \tilde{f}(y), \tilde{f}(x + y))$ is a Heron triple for any $x, y \in G/f^{-1}(0)$. Thus, we may assume that $f^{-1}(0) = \{0\}$ whenever necessary, as we can just replace $G$ with the quotient group $G/f^{-1}(0)$.

    Rearranging the Heron triple condition yields $(f(x + y) - f(x) - f(y))^2 = 4 f(x) f(y)$ for any $x, y \in G$. Thus, for any $x, y \in G$, since $(f(x - y), f(x), f(y))$ and $(f(x + y), f(x), f(y))$ are Heron triples, we get $(f(x + y) - f(x) - f(y))^2 = (f(x - y) - f(x) - f(y))^2 = 4 f(x) f(y)$, and thus
    \[ f(x + y) = f(x - y) \vee f(x + y) + f(x - y) = 2 f(x) + 2 f(y) \quad \forall x, y \in G. \tag{1} \]In particular, this gives $f(2x) \in \{0, 4 f(x)\}$ for any $x \in G$. We now divide into three cases:

    • Case 1. $f(2x) = 4 f(x)$ for all $x \in G$.

      The corresponding answer is the first category shown above. To motivate the solution, you should think that $f$ is supposed to be similar to the square function.

      If $f \equiv 0$ then we are done, so now suppose that $f(g_0) \neq 0$ for some $g_0 \in G$. If $f$ is the square function, then $f(x + y) - f(x) - f(y)$ would be bilinear in terms of $x$ and $y$; it becomes linear if we fix one of $x$ or $y$. Thus it makes sense to guess that $c = 4 f(g_0)$ and $\phi(x) = f(x + g_0) - f(x) - f(g_0)$ works; it already satisfies $c f(x) = \phi(x)^2$ for all $x \in G$, and $c \neq 0$ since $f(g_0) \neq 0$ and $4 = 2^2 \neq 0$ in $R$. It remains to show that $\phi$ is indeed a group homomorphism, which reduces to showing the following equality by substituting $z = g_0$:
      \[ f(x + y + z) + f(x) + f(y) + f(z) = f(x + y) + f(y + z) + f(z + x) \quad \forall x, y, z \in G. \]
      We first show that $f(x + y) + f(x - y) = 2 f(x) + 2 f(y)$ for any $x, y \in G$. By (1), either we are done or we can assume $A = f(x + y) = f(x - y)$. Then $(A, A, f(2x))$ and $(A, A, f(2y))$ are Heron triples, forcing $f(2x) = f(2y) = 4A$. By the case's hypothesis, this implies $f(x) = f(y) = A$. But then $(A, A, A) = (f(x), f(y), f(x + y))$ is a Heron triple, which forces $A = 0$ due to $char(R) \neq 3$. This also yields $f(x + y) + f(x - y) = 2 f(x) + 2 f(y)$, as desired.

      Now the proof of the main equality is as follows: for any $x, y, z \in G$,
      \[ 2 f(x + y + z) + 2 f(x) + 2 f(y) + 2 f(z) = f(2x + y + z) + f(y + z) + f(y + z) + f(y - z) = 2 f(x + y) + 2 f(x + z) + 2 f(y + z). \]The second equality holds since $2x + y + z = (x + y) + (x + z)$ and $y - z = (x + y) - (x + z)$. Now $char(R) \neq 2$ and $R$ is an integral domain, so we can cancel the factor of $2$ out and we are done.

    • Case 2. $f(2x) = 0$ for all $x \in G$.

      The corresponding solutions fall into the second and fourth category. Using the quotient $G/f^{-1}(0)$, we now assume that $f^{-1}(0) = \{0\}$. In particular, the case's hypothesis implies that $G$ is $2$-torsion, or equivalently an $\mathbb{F}_2$-vector space. The claim about what the solutions are is equivalent to saying that $G$ has dimension at most $2$. It now suffices to show that $\dim_{\mathbb{F}_2}(G) \leq 2$, i.e., any $3$ elements of $G$ are linearly dependent over $\mathbb{F}_2$.

      Fix some elements $x, y, z \in G$; the goal is to show that one of $x, y, z$, $x + y, x + z, y + z$, or $x + y + z$ equals zero. Recall that $a^2 + b^2 + c^2 = 2ab + 2bc + 2ca$ can be rearranged to $(c - a - b)^2 = 4ab$. Then $4 f(x) f(w)$ is a square for each $w \in \{x, y, z, x + y, x + z, y + z, x + y + z\}$. For each $\alpha, \beta, \gamma \in \{0, 1\}$ (i.e. $\mathbb{F}_2$), pick some $r_{\alpha \beta \gamma} \in R$ such that $4 f(x) f(\alpha x + \beta y + \gamma z) = r_{\alpha \beta \gamma}^2$. We can change $r_{\alpha \beta \gamma}$ to $-r_{\alpha \beta \gamma}$ if desired.

      Since $(f(x), f(y), f(x + y))$ is Heron, after multiplying by $(4 f(x))^2$, we get that $(r_{100}^2, r_{010}^2, r_{110}^2)$ is also Heron, which forces $r_{110} = r_{100} \pm r_{010}$. The same method yields $r_{\mathbf{v}_1 + \mathbf{v}_2} = r_{\mathbf{v}_1} \pm r_{\mathbf{v}_2}$ for each $\mathbf{v}_1, \mathbf{v}_2 \in \mathbb{F}_2^3$. Fix a sign of $r_{100}$, then choose a sign for $r_{010}$ such that $r_{110} = r_{100} + r_{010}$, and then choose a sign for $r_{001}$ such that $r_{111} = r_{110} + r_{001} = r_{100} + r_{010} + r_{001}$. Now we list some values of $f$:
      \begin{align*}
f(x) &= r_{100}^2 \\
f(y) &= r_{010}^2 \\
f(z) &= r_{001}^2 \\
f(x + y) &= (r_{100} + r_{010})^2 \\
f(x + z) &= (r_{100} \pm r_{001})^2 \\
f(y + z) &= (r_{010} \pm r_{001})^2 \\
f(x + y + z) &= (r_{100} + r_{010} + r_{001})^2
\end{align*}Recall that the goal is to show that at least one of the entries above is equal to $0$. Suppose for the sake of contradiction that none of them are zero. We just case-bash based on the choices of sign in the two $\pm$s.

      First, since $(f(x), f(y + z), f(x + y + z))$ is Heron, we get that $f(y + z)$ is equal to either $(r_{010} + r_{001})^2$ or $(2 r_{100} + r_{010} + r_{001})^2$. In the latter case, some choices of sign of $\pm$ yields $2 r_{100} + r_{010} + r_{001} = \pm r_{010} \pm r_{001}$. If both signs are $-$, then $r_{100} + r_{010} + r_{001} = 0$ and thus $f(x + y + z) = 0$. If both signs are $+$, then $r_{100} = 0$. If one of the sign is $-$, say $2 r_{100} + r_{010} + r_{001} = -r_{010} + r_{001}$, then $r_{100} + r_{010} = 0$ and thus $f(x + y) = 0$; so $f(y + z) = (2 r_{100} + r_{010} + r_{001})^2$ yields a contradiction. The only case remaining is that $f(y + z) = (r_{010} + r_{001})^2$. By symmetry, avoiding contradiction also yields $f(x + z) = (r_{100} + r_{001})^2$.

      Now consider that $(f(x + y), f(x + z), f(y + z))$ is Heron. This implies that, for some choice of the signs $\pm$,
      \[ \pm (r_{100} + r_{010}) \pm (r_{100} + r_{001}) \pm (r_{010} + r_{001}) = 0. \]WLOG suppose that at least two of the signs are $+$. If all three are equal to $+$, then we get $r_{100} + r_{010} + r_{001} = 0$ and thus $f(x + y + z) = 0$. Otherwise, WLOG the signs for $r_{100} + r_{010}$ and $r_{100} + r_{001}$ are $+$. Then the previous equality becomes $2 r_{100} = 0$, which yields that $r_{100} = 0$ and thus $f(x) = 0$; a contradiction as well. All cases yield a contradiction, so one of the seven values shown above must be equal to $0$. We are done.

      (There should be an alternative way, free of the above kind of bash, but either I didn't actually do it or that I forgot how to do it.)

    • Case 3. There exists $g \in G$ such that $f(2g) = 4 f(g) \neq 0$ and $g' \in G$ such that $f(g') \neq 0$ and $f(2g') = 0$.

      Again, pass on to the quotient and WLOG assume that $f^{-1}(0) = \{0\}$. The corresponding solutions fall into the third category; we claim that $g$ is a generator of $G$ of order $4$, which also implies that $g' = 2g$.

      For convenience, let $f(g) = A \neq 0$. Since $2g' = 0$, we have $f(g + g') = f(g - g') = f(g' - g)$. On the other hand, $(f(g + g'), f(g' - g), f(2g) = 4A)$ is a Heron triple, so we necessarily have $f(g + g') = f(g' - g) = A$. Since $f(g') \neq 0$ and $(A, A, f(g')) = (f(g), f(g' - g), f(g'))$ is Heron, we get $f(g') = 4A$. Now both $(f(g' - 2g), f(g' - g), f(g)) = (f(g' - 2g), A, A)$ and $(f(g' - 2g), f(g'), f(2g)) = (f(g' - 2g), 4A, 4A)$ are Heron, so either $f(g' - 2g) = 0$ or $f(g' - 2g) = 4A = 16A$. Since $char(R) \neq 2, 3$, the latter yields $A = 0$; a contradiction. Thus $f(g' - 2g) = 0 \implies g' = 2g$, and so $4g = 2g' = 0$.

      Fix any $x \in G$; the goal is to show that $x \in \{0, g, 2g, -g\}$. By (1), we get either $f(x + 2g) = f(x)$ or $f(x + 2g) + f(x) = 2 f(x + g) + 2 f(g)$. Similarly, either $f(x + 2g) = f(x)$ or $f(x + 2g) + f(x) = 2 f(x - g) + 2 f(g)$ holds, so we have either $f(x + 2g) = f(x)$ or $2 f(x + g) + 2 f(g) = 2 f(x - g) + 2 f(g) \implies f(x + g) = f(x - g)$. It now suffices to show that for any $x \in G$, $f(x + g) = f(x - g)$ implies $x \in \{0, 2g\}$; then we can apply the result to either $x$ or $x + g$.

      Fix any $x \in G$ such that $f(x + g) = f(x - g)$. Since $(f(x + g), f(x - g), f(2g)) = (f(x + g), f(x + g), 4A)$ is Heron and $A \neq 0$, we get $f(x + g) = A = f(g)$. Then we get $f(x + 2g), f(x) \in \{0, 4A\}$. If $f(x + 2g) = f(x) = 4A = f(2g)$, then $(4A, 4A, 4A)$ is a Heron triple; a contradiction, since $4A \neq 0$. So either $f(x + 2g) = 0 \implies x + 2g = 0 \implies x = 2g$, or $f(x) = 0 \implies x = 0$.
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megarnie
5533 posts
#74
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Writing this up again I forgot my solution anyway:

The only solutions are $f\equiv 0$, $f(x) = cx^2$, $f(x) = \begin{cases} 
0 & x\equiv 0\pmod 2 \\
c & x\equiv 1\pmod 2 \\
\end{cases}$, and $f(x) = \frac{n + |n|}{2}  \cdot  c$, for any integer constant $c\ne 0$, where $x_4$ is the remainder when $x$ is divided by $4$ and $n = \nu_2( 2^{x_4} + x_4^3 + x_4^2  + 8 x_4 ) - 1$. We can check that these work, now we prove they are the only solutions.

Let $P(a,b,c)$ be the given assertion.

$P(0,0,0): f(0) = 0$.

Claim: If $a + b + c = 0$ and $f(c) = 0$, then $f(a) = f(b)$.
Proof: We have $f(a)^2 + f(b)^2 = 2f(a)f(b)$, so $(f(a) - f(b))^2 = 0\implies f(a) = f(b)$. $\square$

This implies that $f$ is even.

$P(a, a, -2a): 2f(a)^2 + f(2a)^2 = 2f(a)^2 + 4f(a)f(2a)$, so either $f(2a) = 0$ or $f(2a) = 4f(a)$ for each $a$.

The equation implies that \[ f(a) ^2 + f(b)^2 + f(a+b)^2 = 2f(a)f(b) + 2f(b) f(a+b) + 2f(a+b)f(a) ,\]so \[ f(a+b)^2 - (2(f(a) + f(b))) f(a+b) + (f(a) - f(b))^2 = 0, \]so using the quadratic formula, \[f(a+b) = \frac{ 2f(a) + 2f(b) \pm \sqrt{4 ( (f(a) + f(b))^2  - (f(a) - f(b))^2) }}{2} = \frac{2f(a) + 2f(b) \pm 4\sqrt{f(a)f(b)}}{2}\]
This can be simplified to $f(a) + f(b) \pm 2\sqrt{f(a)f(b)} = \left(\sqrt{f(a)} \pm \sqrt{f(b)}\right)^2$. Let $Q(a,b)$ denote the assertion that \[ f(a+b) = \left(\sqrt{f(a)} \pm \sqrt{f(b)}\right)^2\]
Note that if $f(x) = 0$ for some $x$, then $f(a + x) = f(a)$ by $P(a,x)$, so $f(a) = f( a\pmod x )$ (in fact implying that $f(kx) = 0$ for any integer $k$).

Thus if $f(1) =0$, then $f\equiv 0$. So assume $f(1)  = c \ne 0$.

Now if $f(2) = 0$, we see that $f(2x) = 0$ for any integer $x$, and $f(2x+ 1) = c$ due to $f$ being periodic with period $2$ (from $P(a,2)$). This corresponds to one of the solutions described in the beginning. Now we may assume $f(2) \ne 0$, so $f(2) = 4c$.

If $f(4) = 0$, then $f(4x) = 0$ for any integer $x$, so $f$ is periodic with period $4$. Since $-1 \equiv 3 \pmod 4$ and $f$ is even, we have $f(1\pmod 4) = f(3\pmod 4 )= c$ and $f(2\pmod 4) = 4c$, corresponding to one of the solutions in the beginning.

Now assume $f(1), f(2), f(4)$ are nonzero. We have $f(1) = c, f(2) = 4c, f(4) = 16c$. From $P(2, 1)$, we see that $f(3)$ is either $\left(\sqrt{c} + 2\sqrt{c}\right)^2 = 9c$ or $(\sqrt{c} - 2\sqrt{c})^2 = c$. If $f(3) = c$, from $P(3,1)$, we see that $16c = \left(\sqrt{c} \pm \sqrt{c}\right)^2$, but this means $c = 0$, absurd. Thus, $f(3) = 9c$. Now we induct to show $f(n) = n^2 c $ for any positive integer $n$, with base cases $1,2,3,4$.

Suppose it's true for $1,2,\ldots, n$ and $n\ge 4$. We have from $P(n,1)$ that $f(n+1) = \left( (n \pm 1) \sqrt{c} \right)^2 = (n \pm 1)^2 c $. If $f(n+1) = (n-1)^2 c $, then $P(n-1,2)$ gives that $f(n +1) = \left( (n-1) \sqrt{c} \pm 2 \sqrt{c} \right)^2$, which is either $(n-3)^2 c$ or $(n+1)^2 c$, but this means that $(n-1)^2 = (n-3)^2$ or $(n+1)^2 = (n+3)^2$, both are false for $n > 2$. Thus, we must have $f(n+1) = (n+1)^2 c$, and the induction is complete.

Then of course $f(n) = n^2 c $ for any integer $n$ because $f$ is even.
This post has been edited 3 times. Last edited by megarnie, Jan 28, 2024, 3:40 PM
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naonaoaz
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#75
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Let $P(x,y,z)$ be the assertion. $P(0,0,0) \implies f(0) = 0$. Let $b = 0$ to get
\[f(a)^2+f(-a)^2 = 2f(a)f(-a) \implies (f(a)-f(-a))^2 = 0 \implies f(a) = f(-a)\]for all integers $a$. Next letting $c = -(a+b)$ gives
\[f(a+b)^2 - 2(f(a)+f(b))f(a+b) + (f(a)-f(b))^2 = 0\]\[\implies f(a+b) = f(a)+f(b) \pm 2\sqrt{f(a)f(b)}\]Thus either all $f(x) \ge 0$ or all $\le 0$ or else there will be an complex output of $f$. Since there is a bijection between all positive $f(x)$ and negative, WLOG all $f(x) \ge 0$. Thus
\[\sqrt{f(a+b)} = \left(\sqrt{f(a)}\pm \sqrt{f(b)}\right)^2 \implies g(a+b) = g(a) \pm g(b)\]where $g(x) = \sqrt{f(x)}$. Note that $g(2) = g(1) \pm g(1)$.
  • If $g(2) = 0$, then we can inductively show $g(2k) = 0$ for all integer $k$. Next notice $g(2k+1) = g(2k) \pm g(1) = g(1)$ where we took $+$ sign to avoid $g(2k)$ being negative.
  • If $g(2) = 2g(1)$. Next if $g(4) = 0$, then $g(4k) = 0$ for all integer $k$. Next $g(4k+1) = g(4k) \pm g(1) = g(1)$. Similarly, $g(4k+2) = 2g(1)$. Lastly since $0 = g(1) \pm g(4k+3)$, so $g(4k+3) = g(1)$.

    If $g(4) = 4g(1)$, then $g(3) = 3g(1)$. Next since $g(5) = 4g(1) \pm g(1) = 3g(1) \pm 2g(1)$, we must have $g(5) = 5g(1)$. Induction gives $g(n) = n g(1)$.
Concluding, we see
  • $f(x) = 0$ for even $x$, and $f(x) = c$ for odd $x$.
  • $f(x) = 0$ for $x \equiv 0 \pmod 4$, $f(x) = c$ for $x \equiv 1,3 \pmod 4$, and $f(x) = 4c$ if $x \equiv 2 \pmod 4$.
  • $f(x) = cx^2$ for all $x$.
for any $c \in \mathbb{Z}$.
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AngeloChu
471 posts
#76
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only a simple sketch of my full sol since i got lazy
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KevinYang2.71
394 posts
#77
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We claim the only functions are $\boxed{f(x)\equiv cx^2}$,
\[
\boxed{f(x)\equiv
\begin{cases}
0 & \text{if $x\equiv 0\pmod 2$}\\
c & \text{if $x\equiv 1\pmod 2$}
\end{cases}
},
\]and
\[
\boxed{f(x)\equiv
\begin{cases}
0 & \text{if $x\equiv 0\pmod 4$}\\
c & \text{if $x\equiv 1\pmod 2$}\\
4c & \text{if $x\equiv 2\pmod 4$}
\end{cases}
}
\]for a constant $c\in\mathbb{Z}$. It is not hard to check that these work.

Let $P(a,b,c)$ denote the given assertion. From $P(0,0,0)$ we get $f(0)=0$ and $P(a,-a,0)$ gives us $f(a)=f(-a)$. Then $P(a,b,-a-b)$ yields
\[
f(a)^2+f(b)^2+f(a+b)^2=2f(a)f(b)+2f(a+b)(f(a)+f(b))
\]so
\[
f(a+b)=f(a)+f(b)\pm\sqrt{2f(a)f(b)}\tag{*}
\]for all $a,b\in\mathbb{Z}$. Since $f(a+1)$ is real, it follows that $f(a)$ and $f(1)$ have the same sign. If $f(1)$ is negative we can multiply $(*)$ by $-1$ so $-f$ is positive and we can simply let $c$ be negative in our boxed functions. Thus we may WLOG assume $f(1)\geq 0$. Then $\sqrt{f(a+b)}=\left|\sqrt{f(a)}\pm\sqrt{f(b)}\right|$. Let $g(x):=\sqrt{f(x)}$ so $g(a+b)=|g(a)\pm g(b)|$ for all $a,b\in\mathbb{Z}$. Note that $g(n)=g(-n)\geq 0$ for all $n\in\mathbb{Z}$. Let $r:=g(1)$. Then $g(2)\in\{0,2r\}$.

Case 1: $g(2)=0$. We prove by induction on $|x|$ that
\[
g(x)\equiv
\begin{cases}
0 & \text{if $x\equiv 0\pmod 2$}\\
r & \text{if $x\equiv 1\pmod 2$}
\end{cases}
\]with the base cases trivial. Assume the statement for $|x|\leq k$ (with $k\geq 2$). Then $g(k+1)=|g(k-1)\pm g(2)|=g(k-1)$ so the induction step is complete. This yields the second solution set with $c:=r^2$.

Case 2: $g(2)=2r$. Then $g(3)\in\{r,3r\}$.

Case 2.1: $g(3)=r$. We prove by induction on $|x|$ that
\[
g(x)\equiv
\begin{cases}
0 & \text{if $x\equiv 0\pmod 4$}\\
r & \text{if $x\equiv 1\pmod 2$}\\
2r & \text{if $x\equiv 2\pmod 4$}
\end{cases}
\]with the base cases trivial. Note that $g(4)=|g(3)\pm g(1)|\in\{0,2r\}$ and $g(4)=|g(2)\pm g(2)|\in\{0,4r\}$ so $g(4)=0$. Assume the statement for $|x|\leq k$ (with $k\geq 4$). Then $g(k+1)=|g(k-3)\pm g(4)|=g(k-3)$ so the induction step is complete. This yields the third solution set with $c:=r^2$.

Case 2.2: $g(3)=3r$. We prove by induction on $|x|$ that $g(x)\equiv rx$ with the base cases trivial. Assume the statement for $|x|\leq k$ (with $k\geq 3$). Then $g(k+1)=|g(k)\pm g(1)|=\{c(k-1),c(k+1)\}$ and $g(k+1)=|g(k-1)\pm g(2)|=\{r(k-3),r(k+1)\}$ so $g(k+1)=r(k+1)$. This yields the first solution set with $c:=r^2$.

We are done. $\square$
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ezpotd
1245 posts
#78
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I claim the solution set is given by $f = kx^2$, $f = 0$, $f$ giving distinct values mod $2$, and lastly $f$ giving values mod $4$, where it gives $0$ for $0$, $k$ for $1$, $4k$ for $2$, $k$ for $3$. We check each of these work.

For the first function, we desire to prove $a^4 + b^4 + c^4 = 2a^2b^2 + 2b^2c^2 + 2a^2c^2$, setting $c=  -(a + b)$ and raw expansion finishes.
The second function is obvious.
The third function just requires us to analyze the summation cases mod $2$, of which there are only two, $0 + 0 + 0 =0$ and $0 + 1 + 1 = 0$, both of which obviously work.
The last function requires us to analyze the summation cases mod $4$, of which there are just three, $0 + 0 + 0 = 0, 0 + 1 + 3 = 0, 0 + 2 + 2 = 0$, all of which obviously work.

Set $a,b,c = 0$ to get $f(0) = 0$. Then set $a = -b, c = 0$ to get $f(a) = f(-a)$. We use this without mention, effectively restraining the problem over positive values. Now take $a = b, c = -2a$, and we get $2f(a)^2 + f(-2a)^2 = 2f(a)^2 + 4f(a)f(-2a)$, giving $f(-2a)^2 = 4f(a)f(-2a)$, or $f(2a)(f(2a) - 4f(a)) = 0$. Thus for all $a$, we have $f(2a) = 0$ OR $f(2a) = 4f(a)$. We now divide into two cases.

Case 1: $f(n) = 0$ for some positive $n$. We first inductively prove $f(kn )=  0$ for all $k$. The case of $k = 2$ is obvious. Then take $a = n, b = kn, c = -kn - n$, we get $f(kn + n)^2 = 0$ as desired. Now we take $n | x + y$, setting $a = x ,b = y, c = -n$ gives $f(x)^2 + f(y)^2 = 2f(x)f(y)$, giving $f(x) = f(y)$. This means that if $x \equiv -y \mod n, f(x) = f(y)$ (this also implies that $f$ operates modulo $n$). Now we take the smallest positive root of $f$, $n$, and we claim that all other roots are multiples of $n$. Assume this is not the case, then there must be another root $m$ not divisible by $n$ , and $f(n - (m \mod n))$ must be equal to $f(m)$ and thus equal to $0$, forcing a smaller multiple, contradiction. It suffices now to do casework on $n$. If $n = 1$, we get $f$ is just $0$ over its entire domain. If $n$ has an odd prime divisor, we can find some solution with $k > 0, n \nmid x$ to $2^k x \equiv x \mod n$, which gives $f(2^kx) = 4f(2^{k - 1}x)  = \cdots  = 2^{2k}f(x)$, but since $f$ operates modulo $n$, we must have $f(x) = 2^{2k}f(x)$, which has no solutions since $f(x) \neq 0$. If $n = 2$, the only possible solution is $f$ has some value over odds and another over evens, which always works. If $n = 4$, we must have $f(1) = f(3), f(2) \neq 0$, the latter of which forces $f(2) = 4f(1)$. This constrains the solution to a mod $4$ repetition of $f(0) = 0, f(1) = k, f(2) = 4k, f(3) = k$, which always works. If $n \ge 8$, we have $f(\frac n2) > 0$, so $\frac 14(n^2)f(3) = \frac{1}{16}n^2f(6) = \cdots  f(\frac n2) = \cdots = \frac{1}{16}n^2 f(2) = \frac 14 n^2 f(1)$, so $f(1) = f(3)$, then taking $a = 1, b = 3, c= -4$, gives $f(4) = 4f(1)$, contradiction.

Case 2: $f(n) \neq 0$ for all positive $n$. In this case take $f(1) = k$, now we inductively prove $f(n) = cn^2$ for positive integers, then using $f$ even finishes. The base case of $n = 1,2$ are trivial. After this, take $a = 1, b = n, c=  -n - 1$, this gives $c^2 + c^2n^4 + f(n + 1)^2 = 2c^2n^2 + (2cn^2 + 2c)f(n + 1)$, which gives $f(n + 1) = c(n + 1)^2, c(n  - 1)^2$. If the latter is true, substitute $a = n , b =n , c = -2n$, we then get $f(2n) = 4f(n) = 4cn^2$, but substituting $a = n - 1, b = n+ 1 , c = -2n$ gives $f(2n) = 4f(n - 1) = 4c(n - 1)^2$, contradiction. This forces $f(n + 1) = c(n + 1)^2$.
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pie854
243 posts
#79
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Let $P(a,b,c)$ denote the given assertion. Then, \begin{align*}
P(0,0,0) & \ \Rightarrow 3f(0)^2=6f(0)^2 \Rightarrow f(0)=0 \\
P(a,-a,0) & \ \Rightarrow (f(a)-f(-a))^2=0 \Rightarrow f(a)=f(-a). \end{align*}So $f$ is even, we'll use this fact a lot now. If $f(1)=0$ then $P(n+1,n,-1)$ gives us $f(n+1)=f(n)$ for all $n$ so by induction $f(n)=0$. This clearly works.

Let us now assume $f(1)$ is not $0$, then by scaling lets assume $f(1)=1$. $P(1,1,-2)$ implies that $f(2)\in \{0,4\}$. Let us first consider $f(2)=0$ then it's easy to find $f(n+2)=f(n)$ for all $n$. So $$f (n)=\begin{cases} 0 & n\equiv 0 \pmod 2 \\ 1 & n\equiv 1 \pmod 2\end{cases}.$$It's not too hard to verify by casework that this is indeed a solution.

Let's consider $f(2)=4$. Suppose $f(n)=n^2$ for all $0 \leq n\leq k$ for $k\geq 2$. Then $$P(k,1,-k-1) \Rightarrow f(k+1)^2-2f(k+1)(k^2+1)+(k^2-1)^2=0 \Rightarrow f(k+1)=(k\pm 1)^2.$$Suppose it were that $f(k+1)=(k- 1)^2$. Then $f(k-1)=f(k+1)$ and \begin{align*} P(k+1,k-1,-2k) & \ \Rightarrow f(2k)^2=4f(2k)f(k+1) \\ P(k,k,-2k) & \ \Rightarrow f(2k)^2=4f(2k)f(k) \end{align*}This implies $f(2k)=0$. So, $P(n,2k,-n-2k)$ gives us $f(n+2k)=f(n)$. We have $$P(k-1,k-1,-2k+2)\Rightarrow 2(k-1)^4+2^4=4(k-1)^2\cdot 2^2+2\cdot (k-1)^2\cdot (k-1)^2 \Rightarrow k=0,2 \qquad (\star)$$so $k=2$. So, $f(n+4)=f(n)$ and by induction we get $$f (n)=\begin{cases} 0 & n\equiv 0 \pmod 4 \\ 1 & n\equiv 1,3 \pmod 4 \\ 4 & n\equiv 2 \pmod 4\end{cases}.$$Again, it's not too hard to verify by casework that this is indeed a solution. Now if $f(k+1)=(k+1)^2$ then by induction $f(n)=n^2$ for all $n$, which clearly works. Note that the induction works because $k+1>2$ - look at $ (\star)$.
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cursed_tangent1434
548 posts
#80
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Pretty nasty casework. Note that,
\[P(0,0,0)\implies 3f(0)^2=6f(0)^2\]which gives us that $f(0)=0$. Now, $P(a,-a,-0)$ gives us that
\begin{align*}
    f(a)^2+f(-a)^2 &= 2f(a)f(-a)\\
    f(a)^2-2f(a)f(-a)+f(-a)^2 &= 0\\
    (f(a)-f(-a))^2 &= 0
\end{align*}Thus, $f(-a)=f(a)$ for all $a \in \mathbb{Z}^+$. Now, look at $P(a,b,-a-b)$ to obtain that,
\[f(a)^2+f(b)^2+f(-a-b)^2 =2f(a)f(b)+2f(b)f(-a-b)+2f(-a-b)f(a)\]So,
\begin{align*}
    f(a+b)^2-2f(a+b)(f(a)+f(b))+f(a)^2-2f(a)f(b)+f(b)^2&=0 \\
    (f(a+b)^2-f(a)-f(b))^2 &= 4f(a)f(b)
\end{align*}
Now, note that, this means,
\[(f(2)-2f(1))^2=4f(1)^2\]and thus, $f(2)=4f(1)$ or $f(2)=0$. Based on these two value, we have several cases.

Case 1 : $f(1)=c$ and $f(2)=4c$. In this case note that we have,
\[(f(3)-f(2)-f(1))^2=4f(2)f(1)=16f(1)^2\]so, $f(3)=9c$ or $f(3)=c$. So, we have two sub cases.

\textbf{Subcase 1 : } $f(3)=9c$. Here, simply note that if we have $f(1)=c,\dots, f(n)=n^2c$ for some $n\geq 3$ then,
\begin{align*}
    (f(n+1)-f(n)-f(1))^2 &= 4f(n)f(1)\\
    (f(n+1)-n^2c-c)^2 &= 4n^2c^2
\end{align*}So, $f(n+1)=(n-1)^2c$ or $f(n+1)=(n+2)^2c$. But if $f(n+1)=(n-1)^2c$,
\begin{align*}
    (f(n+1)-f(n-1)+f(2))^2 &= 4f(n-1)f(2)\\
    (f(n+1)-(n-1)^2c+4c)^2 &= 16(n-1)^2c^2\\
    16c^2 &= 16(n-1)^2c^2
\end{align*}which is clearly impossible for all $n\geq 3$. Thus, we must have $f(n+1)=(n+1)^2c$. Thus, by the PMI, we have that for all integers $k \geq 1$, $f(k)=k^2f(1)$. This gives us the function $f(n)=n^2c$ for some constant $c$ for all positive integers $n$.

Subcase 2 : $f(3)=c$. In this case note that we have,
\[(f(4)-8c)^2=(f(4)-2f(2))^2=4f(2)^2=64c^2\]So, $f(4)=16c$ or $f(4)=0$. But note that if $f(4)=16c$ we have
\[4c^2=4f(3)f(1)=(f(4)-f(3)-f(1))^2=(16c-c-c)^2\]so, $4c^2=196c^2$ which requires, $c=0$, which implies that $f(4)=0$ anyways. Thus, in all cases we have that $f(4)=0$.

Note that then, we can obtain
\begin{align*}
    (f(n+4)-f(n)-f(4))^2 &= 4f(n)f(4)\\
    f(n+4)-f(n)-f(4) &= 0 \\
    f(n+4) &= f(n)
\end{align*}for all $n \geq 1$. Thus, the function is periodic with a period of $4$ giving us,
\[
f(n)=
    \begin{cases}
        c & \text{if } n \equiv 1,3 \pmod{4}\\
        4c & \text{if } n \equiv 2 \pmod{4}\\
        0 & \text{if } n \equiv 0 \pmod{4}
    \end{cases}
\]for all $n \in \mathbb{N}$ for some constant $c \in \mathbb{N}$.

Case 2 : $f(2)=0$. Here, note that
\begin{align*}
    (f(n+2)-f(n)-f(2))^2 &= 4f(n)f(2)\\
    f(n+2)-f(n)-f(2) &= 0\\
    f(n+2) &= f(n)
\end{align*}for all $n\geq 1$. Thus, the function is periodic with a period of $2$ giving us the function,
\[
f(n)=
    \begin{cases}
        c & \text{if } n \equiv 1 \pmod{2}\\
        0 & \text{if } n \equiv 0 \pmod{2}
    \end{cases}
\]Thus, we have exhausted all cases and are done.
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AshAuktober
920 posts
#82
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Nice problem!

Show that $f(0) = 0, f(-x) = f(x)$.
Then from here, assuming $f(1) \ne 0$, show that if $f(n) = n^2f(1)$, then $f(n+1) \in \{(n-1)^2f(1), (n+1)^2f(1)\}$.
From here, if $f(n+1) = (n+1)^2f(1) \forall n,$ then $f(x) = f(1)x^2$. Further note that if $f(2) = (1-1)f(1)^2 = 0$ then we get the solution $$f(x) = \begin{cases}
f(1), \quad \text{ if } n \equiv 1 \pmod{2}\\
0, \quad \text{ if } 2 \mid n.
\end{cases}$$so assume otherwise.
If $f(n+1) = (n-1)^2f(1)$ for some $n \in \mathbb{N}_{\ge 2}$, with $f(x) = x^2f(1) \forall x \le n$, comparing the equations of $(2, n-1, a)$ and $(0, n+1,a)$ gives us $f(2)^2 = 4f(2)(n-1)^2f(1)$, or $f(2) \in \{0, 4(n-1)^2f(1)\}$. $f(2) \ne 0$, so $f(2) = 4(n-1)^2f(1)$.
But $4f(1) = f(2) = 4(n-1)^2f(1)  \implies n \in \{0, 2\}$. $n = 0$ is false as $n \in \mathbb{N}_{\ge 2}$. So consider $n = 2$. This is basically the same as $f(3) = f(1)$. Further we may assume $f(2) = 4f(1)$ as previous.
Then $f(4) = f(2\times 2) \in \{0, 16f(1)\},$ and $f(4) = f(3+1) \in \{0, 4f(1)\}$. Since $f(1) \ne 0$, this implies $f(4) = 0$. In particular, $f$ is periodic with period 4, so we get
$$f(x) = \begin{cases}
0, \quad \text{ if } 4 \mid n.\\
f(1) \quad \text{ if } n \equiv 1 \pmod{2}.\\
4f(1) \quad \text{ if } n \equiv 2 \pmod{4}.
\end{cases}$$All of the functions extracted above work, so we're done. $\square$
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Ilikeminecraft
288 posts
#83
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what the impossible p4

The answer is:
\[f\equiv\begin{cases}0&x\equiv0\bmod4\\k&x\equiv1,3\bmod4\\4k&x\equiv2\bmod4\end{cases}\]\[f\equiv\begin{cases}0&x\equiv0\bmod2\\k&x\equiv1\bmod2\end{cases}\]\[f\equiv 0\]
Begin by taking $(0, 0, 0) \implies f(0) = 0, (0, a, -a) \implies f(a)$ is even.

Observe that $(f(a) + f(b) - f(c))^2 = f(a)^2 + f(b)^2 + f(c)^2 - 2f(c)f(a) - 2f(c)f(b) + 2f(a)f(b) = 4f(a)f(b).$ Hence, $f(a)f(b)$ is a perfect square. Let $f \equiv k g^2,$ where $k\in\mathbb Z, g\colon\mathbb Z\to\mathbb N_0$ and $g$ is even. Hence, $g(a)^2 + g(b)^2 - g(a + b)^2 = \pm 2g(a)g(b).$ Hence, $g(a + b) = g(a) + g(b)$ or $g(a + b) = |g(a) - g(b)|.$ Hence, we can assume that $g(1) = 1$.

Now, assume there exists $k$ such that $g(k) = 0.$ Then $g(a) = g(k - a) = 0$ or $g(a) = g(k - a),$ and thus, if $g(k) = 0,$ then $g(a) = g(k - a) = g(a - k).$ Hence, $g$ has period $k.$ However, we also know $g$ is even, and thus, if $k$ is odd, then $g\equiv 0.$ Otherwise, the period is even. If $k= 2, 4,$ then $g\equiv x\pmod2$ or
\[g\equiv\begin{cases}0&x\equiv0\bmod4\\1&x\equiv1,3\bmod4\\2&x\equiv2\bmod4\end{cases}\]Now, assume $k> 4.$ In the original equation, take $(a, a, 2a)$ to get $f(2a) = 4f(a)$ or $f(2a) = 0.$ However, this implies $k \leq 4.$

Now, assume that $g$ has no other roots. Now, we induct. Assume there was a pair $(a, b)$ such that $g(a + b) = |g(a) - g(b)| < g(a) + g(b) = g(a) + g(b - 1) + 1 = g(a + b - 1) + 1,$ which finishes. Thus, $g\equiv x.$ However, this doesn't work.
This post has been edited 1 time. Last edited by Ilikeminecraft, Mar 6, 2025, 2:35 AM
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Nari_Tom
70 posts
#84
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I will write down my solution for some reason.

Indeed, there are only three sets of solutions for this problem. Solutions are:
$(a)$. $f(x)=kx^2$, for all $x \in \mathbb {Z}$, where $k$ is a constant.
$(b)$. $f(x)=0$ for all even $x$, and $f(x)=k$ for all odd $x$ where $k$ is a constant.
$(c)$. $f(x)=k$ for all odd $x$, $f(x)=0$ for all $x \equiv 0 \pmod 4$, $f(x)=4k$ for all $x \equiv 2 \pmod 4$.


It's easy to prove that $f(0)=0$, and $f(x)=f(-x)$.


Lemma (1): If $f(a)=0$, then $f(b)=f(c)$ where $a+b+c=0$.
Proof: $f(a)=0$, then $f(b)^2+f(c)^2=2f(b)f(c)$ $\implies$ $(f(b)-f(c))^2=0$ and result follows.


Lemma (2): $f(2x)[f(2x)-4f(x)]=0$, for all $x \in \mathbb{Z}$.
Proof: Just take $(a=x, b=x, c=-2x)$, then result follows since $f$ is even.


Lemma (3): Let's define $g(x)=\frac {f(x)}{f(1)}$. Then $g(1)=1$. If we have $g(a)=a^2$ and $g(b)=b^2$, then $g(a+b)=(a+b)^2$ or $g(a+b)=(a-b)^2$.
Proof: Just plug $(a=a, b=b, c=-(a+b))$, then we have $[g(a+b)-(a+b)^2]*[g(a+b)-(a-b)^2]=0$, and the result follows.


If $f(1)=0$, then by Lemma (1) we have that $f(a+1)=f(a)$, so our function is const, so $f(x)=0$ in this case.


Otherwise by using Lemma (2) we get that $f(2)=0$ or $f(2)=4f(1)$.


If $f(2)=0$, then by applying Lemma (1) we have that $f(x)=0$ for all even x, and $f(x)=k$ for all odd $x$.


Now assume that $f(2)=4f(1)$. Let's define $g(x)=\frac {f(x)}{f(1)}$. Then $g(1)=1$ and $g(2)=4$. Let $g(3)=d$, then by taking $(a=1, b=2, c=-3)$ we will get that $(a-1)(a-9)=0$. We will consider each cases here.


(Case 1)
Let $g(3)=1$, Then we can get $g(4)=0$ by checking small values. Since $g(4)=4$, by using Lemma (1), we will get $g(n+4)=g(n)$, for all $n$ which gives us Solution(c).


(Case 3)
Let $g(3)=9$, in this case we can easily induct on natural numbers using Lemma (3), which gives us Solution(a).
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