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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.

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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 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
D1027 : Super Schoof
Dattier   1
N 7 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
7 minutes ago
minimizing sum
gggzul   0
9 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
9 minutes ago
0 replies
Prove that 4p-3 is a square - Iran NMO 2005 - Problem1
sororak   23
N 14 minutes ago by reni_wee
Let $n,p>1$ be positive integers and $p$ be prime. We know that $n|p-1$ and $p|n^3-1$. Prove that $4p-3$ is a perfect square.
23 replies
sororak
Sep 21, 2010
reni_wee
14 minutes ago
IMO 2009, Problem 2
orl   142
N 21 minutes ago by pi271828
Source: IMO 2009, Problem 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP = OQ.$

Proposed by Sergei Berlov, Russia
142 replies
orl
Jul 15, 2009
pi271828
21 minutes ago
(a-1)(b-1)(c-1) is a divisor of abc-1
ehsan2004   22
N an hour ago by reni_wee
Source: IMO 1992, Day 1, Problem 1
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1)  \] is a divisor of $abc-1.$
22 replies
ehsan2004
Jan 22, 2005
reni_wee
an hour ago
Balkan MO 2025 p1
Mamadi   0
an hour ago
Source: Balkan MO 2025
An integer \( n > 1 \) is called good if there exists a permutation \( a_1, a_2, \dots, a_n \) of the numbers \( 1, 2, 3, \dots, n \), such that:

\( a_i \) and \( a_{i+1} \) have different parities for every \( 1 \le i \le n - 1 \)

the sum \( a_1 + a_2 + \dots + a_k \) is a quadratic residue modulo \( n \) for every \( 1 \le k \le n \)

Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

Remark: Here an integer \( z \) is considered a quadratic residue modulo \( n \) if there exists an integer \( y \) such that \( y^2 \equiv z \pmod{n} \).
0 replies
1 viewing
Mamadi
an hour ago
0 replies
Number theory
MathsII-enjoy   3
N an hour ago by KevinYang2.71
Prove that when $x^p+y^p$ | $(p^2-1)^n$ with $x,y$ are positive integers and $p$ is prime ($p>3$), we get: $x=y$
3 replies
MathsII-enjoy
Yesterday at 3:22 PM
KevinYang2.71
an hour ago
Inspired by Bet667
sqing   1
N an hour ago by ytChen
Source: Own
Let $ a,b $ be a real numbers such that $a^2+kab+b^2\ge a^3+b^3.$Prove that$$a+b\leq k+2$$Where $ k\geq 0. $
1 reply
sqing
5 hours ago
ytChen
an hour ago
4-var inequality
sqing   1
N 2 hours ago by arqady
Source: SXTB (4)2025 Q2837
Let $ a,b,c,d> 0  $. Prove that
$$   \frac{1}{(3a+1)^4}+ \frac{1}{(3b+1)^4}+\frac{1}{(3c+1)^4}+\frac{1}{(3d+1)^4} \geq \frac{1}{16(3abcd+1)}$$
1 reply
sqing
5 hours ago
arqady
2 hours ago
Extremaly hard inequality
blug   1
N 2 hours ago by arqady
Source: Polish Math Olympiad Training Camp 2024
Let $a, b, c$ be non-negative real numbers. Prove that
$$a+b+c+\sqrt{a^2+b^2+c^2-ab-bc-ca}\geq \sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+c^2}.$$Looking for an algebraic solution!
1 reply
blug
2 hours ago
arqady
2 hours ago
Common external tangents of two circles
a1267ab   56
N 2 hours ago by wassupevery1
Source: USA Winter TST for IMO 2020, Problem 2, by Merlijn Staps
Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$. Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD$ is convex.

Suppose lines $AC$ and $BD$ meet at point $X$, while lines $AD$ and $BC$ meet at point $Y$. Show that $T$, $X$, $Y$ are collinear.

Merlijn Staps
56 replies
a1267ab
Dec 16, 2019
wassupevery1
2 hours ago
That's Vietnamese geo!
wassupevery1   8
N 2 hours ago by cj13609517288
Source: 2025 Vietnam National Olympiad - Problem 3
Let $ABC$ be an acute, scalene triangle with circumcenter $O$, circumcircle $(O)$, orthocenter $H$. Line $AH$ meets $(O)$ again at $D \neq A$. Let $E, F$ be the midpoint of segments $AB, AC$ respectively. The line through $H$ and perpendicular to $HF$ meets line $BC$ at $K$.
a) Line $DK$ meets $(O)$ again at $Y \neq D$. Prove that the intersection of line $BY$ and the perpendicular bisector of $BK$ lies on the circumcircle of triangle $OFY$.
b) The line through $H$ and perpendicular to $HE$ meets line $BC$ at $L$. Line $DL$ meets $(O)$ again at $Z \neq D$. Let $M$ be the intersection of lines $BZ, OE$; $N$ be the intersection of lines $CY, OF$; $P$ be the intersection of lines $BY, CZ$. Let $T$ be the intersection of lines $YZ, MN$ and $d$ be the line through $T$ and perpendicular to $OA$. Prove that $d$ bisects $AP$.
8 replies
wassupevery1
Dec 25, 2024
cj13609517288
2 hours ago
Equation has no integer solution.
Learner94   33
N 2 hours ago by bjump
Source: INMO 2013
Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.
33 replies
Learner94
Feb 3, 2013
bjump
2 hours ago
Problem 1 — Symmetric Squares, Symmetric Products
RockmanEX3   8
N 2 hours ago by Baimukh
Source: 46th Austrian Mathematical Olympiad National Competition Part 1 Problem 1
Let $a$, $b$, $c$, $d$ be positive numbers. Prove that

$$(a^2 + b^2 + c^2 + d^2)^2 \ge (a+b)(b+c)(c+d)(d+a)$$
When does equality hold?

(Georg Anegg)
8 replies
RockmanEX3
Jul 14, 2018
Baimukh
2 hours ago
Problem 2 (First Day)
Valentin Vornicu   83
N May 4, 2025 by Adywastaken
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations

\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]
83 replies
Valentin Vornicu
Jul 12, 2004
Adywastaken
May 4, 2025
Problem 2 (First Day)
G H J
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huashiliao2020
1292 posts
#73
Y by
In OTIS where f(x) is P(x). Notice 0,0,0 yields P(0)=0, whence a,0,0 gives P(a)+P(-a)=2P(a), so P is even. 3x,6x,-2x gives P(-3x)+P(8x)+P(-5x)=2P(7x); It's obvious that for P having degree $2p\ge 6$, the coefficient of the largest degree on the right side is 3^p+8^p+5^p>2*7^p, a contradiction, so our only solutions are of the form P(x)=ax^4+bx^2+c, and checking we need c=0 while all the other coefficients match up. $\blacksquare$
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Shreyasharma
682 posts
#74
Y by
Kinda disappointed this boiled down to guess and check, but okay.

We claim the only solutions are of the form $f(x) = mx^4 + nx^2$, which clearly work.

Let $Q(a, b, c)$ denote the assertion. Clearly $Q(0, 0, 0)$ gives $P(0) = 0$. Then from $Q(x, 0, 0)$ we see $P(x) = P(-x)$ implying $P$ is even. Thus $P$ can be written as $G(x^2) = P(x)$ for some polynomial $G(x)$.

Now we claim that all polynomials of the form $f(x) = mx^4 + nx^2$ work. It is easy to show that these work from just plugging.

Now we claim if $\text{deg}(P) > 6$ fails. To see this we consider the leading coefficients of the polynomial. Note that we want the following to fail,
\begin{align*}
    (a-b)^{2n} + \left(b + \frac{ab}{a+b}\right)^{2n} + \left(a + \frac{ab}{a+b}\right)^{2n} = 2\left(a + b - \frac{ab}{a+b}\right)^{2n}
\end{align*}This is equivalent to showing that the leading coefficients of the following fail,
\begin{align*}
    (a^2 - b^2)^{2n} + \left(2ab + b^2 \right)^{2n} + \left(2ab + a^2 \right)^{2n} = 2\left(a^2 + b^2 + ab\right)^{2n}
\end{align*}Setting $a = 2x$ and $b = x$ we see we need,
\begin{align*}
    (3x)^{2n} + (5x)^{2n} + (8x)^{2n} = 2 \cdot (7x)^{2n}
\end{align*}However it is easy to see the LHS is much greater than the RHS for $n \geq 3$, so it is impossible for this equation to hold. Thus we are done.
This post has been edited 2 times. Last edited by Shreyasharma, Sep 27, 2023, 5:07 AM
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joshualiu315
2534 posts
#75
Y by
Notice $(a,b,c)=(0,0,0)$ gives $P(0)=0$ so then we can plug in $(a,b,c)=(x,0,0)$ to get that $P$ is even.

Set $P(x) = a_{2n}x^{2n} + a_{2n-2}x^{2n-2} + \cdots + a_2x^2 + a_0$. Taking $(a,b,c) = (3x,6x,-2x)$ gives $\deg P <6$. Solving, we get $\boxed{P(x)=cx^4+dx^2, \ c,d \in \mathbb{R}}$.
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abeot
125 posts
#76 • 2 Y
Y by chenghaohu, centslordm
Taking $a=b=c=0$ we find that $P(0) = 0$.
Taking $a=b=0$ we find that $P(-c) = P(c)$, so $P$ must be even.

Now, let $a = -2k$, $b = 3k$ and $c = 6k$.
Clearly this satisfies $ab+bc+ca=0$, and we obtain
\[ P(5k) + P(3k) + P(8k) = 2P(7k) \]Now, suppose that $P(x)$ has degree $n$. Then we obtain
\[ 5^n + 3^n + 8^n = 2\cdot 7^n \]From Bernoulli we see that $\frac{8^7}{7^7} \geq 2$. Thus, $n \leq 7$. Taking modulo 6, we see that $n \neq 6$.
Therefore, the maximal degree is $4$, as $n$ must be even.

Then, we find that $P(x) = k_1x^4 + k_2x^2$ for some reals $k_1$ and $k_2$. To see how $x^4$ works, note that
\[ (a-b)^4 + (b-c)^4 + (c-a)^4 - 2(a+b+c)^4 = -6(ab+bc+ca)(ab+bc+ca+a^2+b^2+c^2) \]and $x^2$ works as
\[ (a-b)^2 + (b-c)^2 + (c-a)^2 - 2(a+b+c)^2 = -6(ab+bc+ca) \]so indeed all polynomials of that form work. $\blacksquare$
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jolynefag
125 posts
#77
Y by
Let our polynomial be of the form $P(x)=a_n \cdot x^n+a_{n-1} \cdot x^{n-1}+...+a_1\cdot x+a_0$. Let's play with the coefficients of the polynomial, that is, if there is some $c_i*A$ on the left in the expression, where say $A$ is an expression of $a,b,c$, then we should have the same thing on the right.

Let's look at how the coefficient $a_0$ behaves: $$a_0+a_0+a_0=2a_0,$$
$$\Rightarrow \a_0=0.$$

Let's see how $a_1$ behaves: $$a_1((a-b)+(b-c)+(c-a))=2a_1(a+b+c),$$$$\Rightarrow 0=a_1(a+b+c).$$
$a+b+c$ is not always $0$, so $a_1=0$.

And how $a_2$ behaves: $$a_2((a-b)^2+(b-c)^2+(c-a)^2)=2a_2(a+b+c)^2,$$$$\Rightarrow a_2(2a^2+2b^2+2c^2)=a_2(2a^2+2b^2+2c^2).$$
Everything works, so we can say that $a_2$ can essentially be anything, and we leave it that way.
Of course, I would like to work with $a_3$ further, but I'm so lazy, a lot of numbers appear with it, and something tells me that equality does not always happen, I thought. Then, it would be worthwhile to come up with some universal values for $a,b,c$ so that the condition $ab+bc+ac=0$ would be satisfied at the same time, and so that the coefficients could be conveniently estimated.

The equality of the coefficient $a_i$ looks like this: $$a_i((a-b)^i+(b-c)^i+(c-a)^i)=2a_i(a+b+c)^i.$$It is very inconvenient to disclose this whole thing with large values of $i$, then let's find convenient values for $a,b,c.$ Then, let's look at $(a-b)$, I thought it would be quite convenient if $a=b+1$ and the value would be identical to $1$ for any $i$. With the same logic, I made it so that $c=b+2$, then let's see what happens to $ab+bc+ac$:

$$b(b+1)+b(b+2)+(b+1)(b+2)=0,$$$$3b^2+6b+2=0,$$$$ \Rightarrow b_1=\frac{1}{\sqrt{3}}-1, \ b_2=-\frac{1}{\sqrt{3}}-1.$$We get two roots, I see that the first one is more convenient and I take it. Let's look at our coefficient $a_i$ under these values: $$a_i((\frac{1}{\sqrt{3}}-(\frac{1}{\sqrt{3}}-1))^i+((\frac{1}{\sqrt{3}}-1)-(\frac{1}{\sqrt{3}}+1))^i+((\frac{1}{\sqrt{3}}+1)-\frac{1}{\sqrt{3}})^i)=2a_i(\sqrt{3})^i,$$$$\Rightarrow a_i((-2)^i+2)=2a_i(\sqrt{3})^i.$$Let's say $a_i$ is not $0$, then $(-2)^i+2=2(\sqrt{3})^i.$ With $i\geq 3$ and odd, the two sides will have different signs, and with $i\geq 6$ and $i$ even, the growth will be faster on the left than on the right (it can be proved by induction that there will always be more on the left with $i\geq 6$.) Works for $i=2.4$, but this does not mean that this will always be the case. If $i=2$ we have checked. Check $i=4$, use the facts that $(a+b+c)^2=a^2+b^2+c^2$, $(ab+bc+ac)^2=0,$ $(a^2+b^2+c^2)(ab+bc+ac)=0$ and make sure that the equality is identical.

Accordingly, any coefficient except $a_4$ and $a_2$ must be zero, and these two can take any value.

Answer: $\boxed{P(x)=a_4\cdot x^4+a_2 \cdot x^2.}$
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kamatadu
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#78 • 1 Y
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$P \equiv kx^4 + lx^2$ works where $k$, $l\in \mathbb R$.

Let $Q(a,b,c)$ denote the assertion $P(a-b)+P(b-c)+P(c-a) = 2P(a+b+c)$.

$Q(0,0,0)\implies P(0) = 0$.

\begin{align*}
    Q\left(c,c,-\frac{c}{2}\right) &\implies P(0) + P\left(\frac{3c}{2}\right) + P\left(\frac{-3c}{2}\right) = 2 P\left(\frac{3c}{2}\right)\\
    &\implies P\left(\frac{-3c}{2}\right)=P\left(\frac{3c}{2}\right)\\
    &\implies P(-x) = P(x)\\
    &\implies P(x) \text{ is even}
.\end{align*}
All the terms are of even degree.

$P(x) = a_nx^{2n} + \cdots + a_1x^2 + a_0x^0$. So $P(0) = 0 \implies a_0 = 0$.

Now $Q\left(\frac{-c}{3},\frac{c}{2},c\right) \implies P\left(\frac{-5c}{6}\right) + P\left(\frac{-c}{2}\right) + P\left(\frac{4c}{3}\right) = 2P\left(\frac{7c}{6}\right)$.

So, we have,
\[  P\left(\frac{5c}{6}\right) + P\left(\frac{c}{2}\right) + P\left(\frac{4c}{3}\right) = 2P\left(\frac{7c}{6}\right). \]
Now comparing the coefficients of the $c^{2i}$ term, we get that,
\[ a_i\left(\frac{5}{6}\right)^{2i} + a_i\left(\frac{1}{2}\right)^{2i} + a_i\left(\frac{4}{3}\right)^{2i} = 2\left(\frac{7}{6}\right)^{2i}. \]$\implies a_i(5^{2i} + 3^{2i} + 8^{2i}) = a_i(2\cdot 7^2i)$.

If $a_i\neq 0$, then $25^i + 9^i + 64^i = 2\cdot 49^i$. Now note that $64^i > 2\cdot 49^i$ for all $i\ge 3$.

Thus we must have $i\le 2$. Checking gives that $i=1$ and $2$ both work. So we get our desired form and we are done. :yoda:
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shendrew7
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#79
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We claim our solutions are of the form $\boxed{P(x)=px^4+qx^2, \quad p,q \in \mathbb{R}}$, which can be tested to work. Denote the assertion as $A(a,b,c)$. Then
\begin{align*}
A(0,0,0) &\implies P(0)=0 \\
A(a,0,0) &\implies P(a)=P(-a) \implies P \text{ even.} \\
A(1, 1-\sqrt 3, 1+\sqrt 3) &\implies (\sqrt 3)^{2k} + (-2\sqrt 3)^{2k} + (\sqrt 3)^{2k} = 2 \cdot 3^{2k} \\
&\implies 4^k+2 = 2 \cdot 3^k \\
&\implies k=1,2.
\end{align*}
Hence the only possible degrees of nonzero terms of $P$ are 2 and 4, as desired. $\blacksquare$
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Cusofay
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We can check that $P(x)=mx^4+nx^2$ for real numbers $m,n$ works and we want to prove its uniqueness.
Note that if $P$ and $Q$ are solutions, $aP+bQ$ is a solution for all real numbers $a,b$ so it's enough to find all solutions of the form $x^n$.$P(0,0,0)$ and $P(0,0,x)$ gives us that $n$ is even

$$P(6,3,-2) \Rightarrow 3^n+5^n+(-8)^n=2 \cdot 7^n$$
If $n=0$, then this obviously doesn't hold.

If $n=2$, then $9+25+64= 98 =2 \cdot 49$, which works.

If $n=4$, then $81+625+4096 = 4802 = 2 \cdot 2401 $, which also works.


If $n=6$, then $$3^6+5^6+8^6>8^6=(7+1)^6=7^6+\frac{6}{7} \cdot 7^6 + \frac{15}{49} \cdot 7^6 = (2+\frac{8}{49}) \cdot 7^6>2 \cdot 7^6,$$which is a contradiction.

If $n \ge 8$, then $$3^n+5^n+8^n>8^n=(7+1)^n>7^n+n \cdot 7^{n-1} > 2 \cdot 7^n,$$contradiction.

$$\mathbb{Q.E.D.}$$
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john0512
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#81
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We claim that the answer is $P(x)=rx^4+sx^2$ for any real number $r,s$, which clearly works.

By plugging $a=b=c=0$, we find that $P(0)=0$. Then, plugging $b=c=0$, we have that $P$ is even.

We would like to cancel the right side by setting $a+b+c=0$, but this isn't possible since $a+b+c=0,ab+ac+bc=0$ implies $a=b=c=0$ over reals. The next best thing would be to consolidate terms of the left hand side: $$a-b=b-c$$$$c=2b-a.$$Plugging $c=2b-a$ gives us that $$2P(a-b)+P(2b-2a)=2P(3b).$$However, we also have $$ab+ac+bc=0$$$$ab+(2b-a)(a+b)=0$$$$a^2-2ab-2b^2=0$$$$a=b(1\pm \sqrt{3}).$$We will only use $a=b(1+\sqrt{3}),$ which when plugged into $$2P(a-b)+P(2b-2a)=2P(3b),$$we get $$2P(b\sqrt{3})+P(-2b\sqrt{3})=2P(3b).$$We will equate coefficients of $b^{2n}$ on both sides, noting that $P$ is even. If the coefficient of $x^{2n}$ in $P$ is $a_2n$, then we have $$a_{2n}(2\cdot 3^n+12^n)=a_{2n}(2\cdot 9^n).$$Thus, or each nonnegative integer $n$, either $$a_{2n}=0$$or $$2\cdot 3^n+12^n=2\cdot 9^n.$$This is only true for $n=1$ and $n=2$, as we can test that $n=0$ doesn't work, and $12^n>2\cdot 9^n$ for $n\geq 3.$ Thus, only the $x^2$ and $x^4$ coefficients are allowed to be nonzero, thus done.

remark: The main idea here is to make a substitution that, combined with the constraint of $ab+ac+bc=0$, makes it so that all terms are of the form $$kP(rb).$$This is very nice because this means that the $x^n$ coefficent of this only depends on the $x^n$ coefficient of $P$, allowing us to easily equate coefficients. By only setting "homogeneous" constraints, like $a-b=b-c$ that I used, one can prevent the input of the polynomial from having a constant shift. In fact, $$(a,b,c)=(b(1+\sqrt{3}),b,b(1-\sqrt{3}))$$is really the only substitution you need to solve this problem.
This post has been edited 1 time. Last edited by john0512, Feb 10, 2024, 4:36 AM
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john0512
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Cusofay wrote:
We can check that $P(x)=mx^4+nx^2$ for real numbers $m,n$ works and we want to prove its uniqueness.
Note that if $P$ and $Q$ are solutions, $aP+bQ$ is a solution for all real numbers $a,b$ so it's enough to find all solutions of the form $x^n$.$P(0,0,0)$ and $P(0,0,x)$ gives us that $n$ is even

$$P(6,3,-2) \Rightarrow 3^n+5^n+(-8)^n=2 \cdot 7^n$$
If $n=0$, then this obviously doesn't hold.

If $n=2$, then $9+25+64= 98 =2 \cdot 49$, which works.

If $n=4$, then $81+625+4096 = 4802 = 2 \cdot 2401 $, which also works.


If $n=6$, then $$3^6+5^6+8^6>8^6=(7+1)^6=7^6+\frac{6}{7} \cdot 7^6 + \frac{15}{49} \cdot 7^6 = (2+\frac{8}{49}) \cdot 7^6>2 \cdot 7^6,$$which is a contradiction.

If $n \ge 8$, then $$3^n+5^n+8^n>8^n=(7+1)^n>7^n+n \cdot 7^{n-1} > 2 \cdot 7^n,$$contradiction.

$$\mathbb{Q.E.D.}$$

unfortunately, the linear combination of any two solutions also being a solution does not necessarily imply that checking $x^n$ suffices

for example, if hypothetically the solution set is $r(x+1)$ for real numbers $r$, then it is true that the linear combination of any two solutions is still a solution, but in this case there are no solutions of the form $x^n$ even though there does exist solutions.

the issue here is that $1,x,x^2,\dots$ is not the only basis for real polynomials, and there are plenty of other choices, so closure under addition like this does not imply that only checking these suffices

it's ok though :D I wish you good luck in the future!
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pie854
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#83
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The answer is $f(x)=ux^4+vx^2$ where $u,v\in \mathbb R$. This polynomial indeed satisfies the problem.

We get from the equation that $f(0)=0$ and $f(x)=f(-x)$ for all $x\in \mathbb R$. So we can write $$f(x)=\sum_{k=1}^n a_k x^{2k}$$where $a_n\neq 0$. Now we put $(a,b,c)=\left(x,-\frac{x}{1+x},1\right)$ into the equation and after clearing the denominator we get $$\sum_{k=1}^n a_k Q_k(x)=0\qquad(\star)$$for all $x\in \mathbb R$, where $$Q_k(x)=x^{2k}(x+2)^{2k}+(2x+1)^{2k}+(x^2-1)^{2k}-2(x^2+x+1)^{2k}.$$We can check that $Q_2(x)=Q_4(x)=0$ for all $x$. And for $k>2$ we can prove via the multinomial theorem that $\deg Q_k=4k-2$.

Now we claim that $n\leq 2$, which will finish the problem. Suppose ftsoc that $n>2$ and let $\deg Q_n=m$. Since $m>\max_{1\leq k\leq n-1} \deg Q_k$, it follows that the coefficient of $x^m$ in $(\star)$ is some nonzero number times $a_n$. Thus we must have $a_n=0$, which is the desired contradiction.
This post has been edited 1 time. Last edited by pie854, May 19, 2024, 4:41 AM
Reason: changed P to f
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sansgankrsngupta
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#84
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Am I the only one to find this problem very easy for a P2 or infact even a P1?
This post has been edited 1 time. Last edited by sansgankrsngupta, Nov 29, 2024, 12:54 PM
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Ilikeminecraft
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Let $S$ be the set of solutions. We prove that $S = \{rx^2 + sx^4 \mid|r, s\in\mathbb R\}$

$(0, 0, 0) \implies P(0) = 0, (0, 0, c) \implies P(c)= P(-c),$ so $P$ is an even function.
Claim: $S$ is a $\mathbb R$-vector space
Proof: This is obvious.
Claim:$P(x) = x^2, x^4$ both work.
Proof: Both are just expansion
Claim: $\deg P \leq 4$
Proof: Take $(6x, 3x, -2x)$ to get $P(3x) + P(5x) + P(8x) = 2P(7x).$ Hence, if a polynomial works, then we require $3^{2n} + 5^{2n} + 8^{2n} = 2\cdot7^{2n}$ where $n = \deg P.$ However, simply note that $8^{2n}\geq2\cdot7^{2n}$ for $n\geq3.$
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Ihatecombin
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#86
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We claim the only polynomials which work are those of the form \(f(x) = ax^4 + bx^2\). It can be shown (painfully) that \(f(x) = x^4\) works, thus \(P(x) = ax^4\) works.
Similarly it can be shown that \(bx^2\) works, thus \(f(x) = ax^4 + bx^2\) works, we shall show that this is the only solution.
Claim 1: \(f(0) = 0\) and \(f(x)\) is even
Proof:
We can just substitute \(b = c = 0\), we immediately obtain
\[f(a) + f(0) + f(-a) = 2f(a) \Longrightarrow f(a) = f(-a) + f(0)\]Since the constant terms of both sides of the equation have to be equal, it follows that \(f(0) = 0\), thus \(f(x)\) is even. $\blacksquare$

The main idea is that in the equation
\[f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c)\]if \(f(x) = a_{n}x^n + a_{n-1}x^{n-1} + \cdots + a_{0}\) works, then \(f(n) = x^n\), \(f(n) = x^{n-1}\), etc, also has to work. Thus by claim \(1\), we simply need to show that \(f(x) = x^{2k}\) fails for \(k > 2\). Let us first show that \(x^{2k}\) fails
Claim 2: \(f(x) = x^{2k}\) fails for \(k > 2\)
Proof:
We can substitute \(c = \frac{-ab}{a+b}\) to get rid of the algebraic condition, thus our equation becomes
\[{(a-b)}^{2k} + {\left(b + \frac{ab}{a+b}\right)}^{2k} + {\left(\frac{-ab}{a+b} - a\right)}^{2k} = 2{\left(a+b-\frac{ab}{a+b}\right)}^{2k}\]We can multiply both sides by \({(a+b)}^{2k}\) to obtain
\[{(a^2-b^2)}^{2k} + {(b^2 + 2ab)}^{2k} + {(a^2 + 2ab)}^{2k} = 2{(a^2 + ab + b^2)}^{2k}\]Now we can substitute \(a = 2\) and \(b = 1\) which gives
\[3^{2k} + 5^{2k} + 8^{2k} = 2 \cdot 7^{2k} \Longrightarrow {\left(\frac{3}{7}\right)}^{2k} + {\left(\frac{5}{7}\right)}^{2k} + {\left(\frac{8}{7}\right)}^{2k} = 2\]Since \({\left(1+\frac{1}{7}\right)}^{6} > 2\) by the binomial theorem, we are done. $\blacksquare$

Now we need to show that if \(f(x) = a_{n}x^n + a_{n-1}x^{n-1} + \cdots + a_{0}\) works, then \(f(n) = x^n\), \(f(n) = x^{n-1}\), etc, also has to work.
Assume \(f(x)\) works for all \((a,b,c)\) with \(ab+bc+ca = 0\) however there exists some \(m\) such that \(a_{m} \neq 0\), but \(f(x) = x^{m}\) fails.

Let \(k\) denote the largest value such that \(f(x) = a_{n}x^n + a_{n-1}x^{n-1} + \cdots + a_{k}x^{k} + \cdots + a_{0}\) works, but \(f(x) = x^k\) fails.
Due to the maximality of \(k\), it follows that if \(a_{m} \neq 0\) and \(m > k\) then \(x^m\) works, thus we can just get rid of these terms and only look at the function
\(g(x) = a_{k}x^{k} + a_{k-1}x^{k-1} + \cdots + a_{0}\), we also preemptively define \(h(x) = g(x) - a_{k}x^k = a_{k-1}x^{k-1} + \cdots + a_{0}\)

Let \((p,q,r)\) denote some triple of values such that \(pq+qr+rp = 0\) and for which
\[{(p-q)}^{k} + {(q-r)}^{k} + {(r-q)}^{k} \neq 2{(p+q+r)}^{k}\]such a triple clearly exists since we assumed that \(f(x) = x^k\) doesn't fulfill the functional equation.
Let \({(p-q)}^{k} + {(q-r)}^{k} + {(r-q)}^{k} - 2{(p+q+r)}^{k} = \delta\), since we have assumed that \(g(x)\) works, it follows that
\[g(p-q) + g(q-r) + g(r-p) = 2g(p+q+r)\]Thus
\[a_{k} \cdot \delta = 2h(p+q+r) -  h(p-q) - h(q-r) - h(r-p)\]Notice that if the functional equation holds for some \((p,q,r)\), it also holds for some \(cp,cq,cr\) where \(c\) is an arbitrary constant. Thus by scaling the variables by \(c\) we have
\[a_{k} \cdot \delta \cdot c^{k} = 2h(cp+cq+cr) - h(cp-cq) - h(cq-cr) - h(cr-cp)\]Since \(h(x)\) is a degree \(k-1\) polynomial at best, it follows that the right hand side grows by at most \(c^{k-1}\), whereas the left hand side grows by \(c^{k}\).
Thus for sufficiently large \(c\), the left hand side cannot possibly equal the right hand side. Thus we are done.
This post has been edited 1 time. Last edited by Ihatecombin, Apr 16, 2025, 2:47 PM
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Adywastaken
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#87
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We first proceed similarly to other solutions.
$a=b=c=0\implies P(0)=0$
$a=b=0\implies P(-c)=P(c)$
$(a, b, c)=(6x, 3x, -2x)$. Then, if $\deg(P)=n$,
$3^n+5^n+8^n=2\cdot7^n$
$n=4$ works since $\frac{4802}{2401}=2$, and there are no other solutions.
Since $P$ is even and of degree 4, let $P(x)=ax^4+bx^2+c$. Plugging in, $c=0$.
So, $P(x)=ax^4+bx^2$.
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