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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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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
Functional equation of nonzero reals
proglote   8
N 14 minutes ago by jasperE3
Source: Brazil MO 2013, problem #3
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
8 replies
proglote
Oct 24, 2013
jasperE3
14 minutes ago
Interesting inequalities
sqing   1
N 26 minutes ago by sqing
Source: Own
Let $ a,b\geq  0 ,a^2-ab+b^2+a+b=3  $. Prove that
$$  \frac{39+\sqrt{13}}{78}\geq  \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq  \frac{1}{2}$$Let $ a,b\geq  0 ,a^2+ab+b^2+a+b=3  $. Prove that
$$  \frac{19+\sqrt{10}}{39}\geq  \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq    \frac{39+\sqrt{13}}{78}$$Let $ a,b\geq  0 ,a^2+ab+b^2+a+b=5  $. Prove that
$$  \frac{3}{5}> \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq     \frac{185+3\sqrt{21}}{402}$$
1 reply
+1 w
sqing
an hour ago
sqing
26 minutes ago
4-var inequality
sqing   2
N an hour ago by sqing
Source: SXTB (4)2025 Q2837
Let $ a,b,c,d> 0  $. Prove that
$$   \frac{1}{(3a+1)^4}+ \frac{1}{(3b+1)^4}+\frac{1}{(3c+1)^4}+\frac{1}{(3d+1)^4} \geq \frac{1}{16(3abcd+1)}$$
2 replies
sqing
Yesterday at 2:59 PM
sqing
an hour ago
Inspired by Bet667
sqing   2
N an hour ago by sqing
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. $
2 replies
sqing
Yesterday at 2:46 PM
sqing
an hour ago
find positive n so that exists prime p with p^n-(p-1)^n$ a power of 3
parmenides51   14
N 2 hours ago by MathLuis
Source: JBMO Shortlist 2017 NT5
Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$.

Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.
14 replies
parmenides51
Jul 25, 2018
MathLuis
2 hours ago
GCD of terms in a sequence
BBNoDollar   1
N 2 hours ago by mashumaro
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
1 reply
BBNoDollar
6 hours ago
mashumaro
2 hours ago
Diodes and usamons
v_Enhance   46
N 2 hours ago by HamstPan38825
Source: USA December TST for the 56th IMO, by Linus Hamilton
A physicist encounters $2015$ atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon $A$ to any other usamon $B$. (This connection is directed.) When she does so, if usamon $A$ has an electron and usamon $B$ does not, then the electron jumps from $A$ to $B$. In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible?

Proposed by Linus Hamilton
46 replies
v_Enhance
Dec 17, 2014
HamstPan38825
2 hours ago
China South East Mathematical Olympiad 2013 problem 2
s372102   3
N 2 hours ago by AGCN
$\triangle ABC$, $AB>AC$. the incircle $I$ of $\triangle ABC$ meet $BC$ at point $D$, $AD$ meet $I$ again at $E$. $EP$ is a tangent of $I$, and $EP$ meet the extension line of $BC$ at $P$. $CF\parallel PE$, $CF\cap AD=F$. the line $BF$ meet $I$ at $M,N$, point $M$ is on the line segment $BF$, the line segment $PM$ meet $I$ again at $Q$. Show that $\angle ENP=\angle ENQ$
3 replies
s372102
Aug 10, 2013
AGCN
2 hours ago
f(x+y) = max(f(x), y) + min(f(y), x)
Zhero   50
N 2 hours ago by lpieleanu
Source: ELMO Shortlist 2010, A3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.

George Xing.
50 replies
Zhero
Jul 5, 2012
lpieleanu
2 hours ago
Similar to iran 1996
GreekIdiot   1
N 2 hours ago by Lufin
Let $f: \mathbb R \to \mathbb R$ be a function such that $f(f(x)+y)=f(f(x)-y)+4f(x)y \: \forall x,y \: \in \: \mathbb R$. Find all such $f$.
1 reply
GreekIdiot
Apr 26, 2025
Lufin
2 hours ago
Prove that 4p-3 is a square - Iran NMO 2005 - Problem1
sororak   24
N 3 hours ago by EmersonSoriano
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.
24 replies
sororak
Sep 21, 2010
EmersonSoriano
3 hours ago
IMO ShortList 1999, combinatorics problem 4
orl   27
N 3 hours ago by cursed_tangent1434
Source: IMO ShortList 1999, combinatorics problem 4
Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.
27 replies
orl
Nov 14, 2004
cursed_tangent1434
3 hours ago
Geometry
Lukariman   2
N 3 hours ago by Lukariman
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that <HDM = 2∠AMP.
2 replies
Lukariman
Yesterday at 12:43 PM
Lukariman
3 hours ago
IMO 2010 Problem 1
canada   119
N 3 hours ago by lpieleanu
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$

Proposed by Pierre Bornsztein, France
119 replies
canada
Jul 7, 2010
lpieleanu
3 hours ago
IMO ShortList 1999, number theory problem 1
orl   62
N Apr 26, 2025 by Ilikeminecraft
Source: IMO ShortList 1999, number theory problem 1
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.
62 replies
orl
Nov 13, 2004
Ilikeminecraft
Apr 26, 2025
IMO ShortList 1999, number theory problem 1
G H J
Source: IMO ShortList 1999, number theory problem 1
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orl
3647 posts
#1 • 7 Y
Y by ahmedosama, Davi-8191, Adventure10, megarnie, HWenslawski, Mango247, cubres
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.
Attachments:
This post has been edited 2 times. Last edited by orl, Nov 14, 2004, 10:16 PM
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grobber
7849 posts
#2 • 11 Y
Y by feridverdiyev, alreva99, Adventure10, megarnie, ngocthinht1k10ltt, Vladimir_Djurica, Mai-san, Mango247, Stuffybear, cubres, and 1 other user
First of all, if $p=2$, then $x$ is $1$ or $2$. Assume now $p\ge 3$.

Clearly, $x$ must be odd (let's say that $x=2k+1$), and let's assume $x>1$. Let $q$ be the smallest prime divisor of $x$. We have $q|((p-1)^2x-1,(p-1)^{q-1}-1)=(p-1)^2-1=p(p-2)$. Assume $q\ne p$. We find that $q|p-2$. At the same time, we have $q|(p-1)^x+1=(p-1)^{2k+1}+1=[(p-1)^{2k}-1](p-1)+p$, so $q|p$, which contradicts the fact that $q|p-2$ and $q$ is odd. This means that $q=p$, so $x\in\{p,2p\}$, and since it's odd, we have $x=p$.

We now have to find those odd primes $p$ s.t. $p^{p-1}|(p-1)^p+1$. After expanding the binomial on the right, we find that the expression on the right is $p^2+kp^3$, so $p-1\le 2$, meaning that the only odd prime satisfying this is $p=3$ (it it does satisfy the relation, since $3^2|2^3+1$). The solutions are then $(1,p)$ for any prime $p$, $(2,2)$ and $(3,3)$.
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orl
3647 posts
#3 • 3 Y
Y by Adventure10, Mango247, cubres
OFFICIAL SOLUTION

Clearly we have the solutions $(1,p)$ and $(2,2)$, and for every other solution $p \geq 3$. It remains to find the solutions $(x,p)$ with $x \geq 2$ and $p \geq 3$. We claim that in this case $x$ is divisible by p and $x y 2p$, whence $x = p$. This will lead to

\[p^{p-1}|(p-1)^{p}+1=p^{2}\left(p^{p-2}-{p \choose 1}p^{p-3}+
\cdots + -{p \choose p-3}p-{p \choose p-2}+1\right)\]

therefore, because all the terms in the brackets excepting the last one is divisible by $p$,$p-1 \leq 2$. This leaves only $p=3$ and $x=3$. Let us prove now the claim. Since $(p-1)^{x}+1$ is odd, so is $x$ (therefore $x < 2p$). Denote by q the smallest prime divisor of $x$. $q|(p-1)^{x}+1$ we get $(p-1)^{x} \equiv -1 \pmod {q}$ and $(q,p-q)=1$. But $(x,p-q)=1$ (from the choice of q) leads to the existence of integers $u,v$ such that $ux+v(q-1)=1$, whence $p-1 \equiv (p-1)^{ux}$. $(p-1)^{v(q-1)} \equiv (-1)^{u} \cdot 1^{v} \equiv -1 \pmod {q}$, because $u$ must be odd. This shows that $q|p$, therefore $q=p$. In conclusion the required solutions are $(2,2), (3,3)$ and $(1,p)$, where $p$ is an arbitrary prime.
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Davron
484 posts
#4 • 3 Y
Y by Adventure10, Mango247, cubres
@all : can you please explain :-)

$q|[(p-1)^2x-1,(p-1)^{q-1}-1)=(p-1)^2-1$

Sincerely Davron
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marko avila
521 posts
#5 • 5 Y
Y by Adventure10, Mango247, Mango247, Mango247, cubres
grobber wrote:
At the same time, we have $q|(p-1)^{x}+1=(p-1)^{2k+1}+1=[(p-1)^{2k}-1](p-1)+p$, so $q|p$, which contradicts the fact that $q|p-2$ and $q$ is odd.
i dont get this why does q divide p. please someone explain.
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Mathias_DK
1312 posts
#6 • 2 Y
Y by Adventure10, cubres
orl wrote:
Find all the pairs of positive integers $ (x,p)$ such that p is a prime, $ x \leq 2p$ and $ x^{p - 1}$ is a divisor of $ (p - 1)^{x} + 1$.
First the case where $ p=2$: $ x \mid 2$. So there are solutions $ (p,x) = (2,1)$ and $ (p,x) = (2,2)$.
Then the case where $ x=1$. (I will use that $ x$ has prime divisors later) $ 1^{p-1} \mid (p-1)^1 + 1 \iff 1 \mid p$ always holds. So $ (1,p)$ are also solutions.
$ x^{p-1} \mid (p-1)^x + 1 \mid (p-1)^{2x} - 1$. Let $ q$ be the smallest primedivisor of $ x$. Then $ (p-1)^{2x} \equiv 1 \bmod q \Rightarrow (p-1)^{(2x,q-1)} \equiv 1 \bmod q$. But $ (2x,q-1) \le 2(x,q-1) = 2$. So $ q \mid (p-1)^2-1 = p(p-2)$.
If $ q \mid p$ then $ q=p$. Since $ x$ is odd and $ x \leq 2p$ we see that $ x=p > 2$. Then $ p^{p-1} \mid (p-1)^p + 1 = (p-1)^p - (-1)^p$
It is well known that $ v_p(a^n-b^n) = v_p(n) + v_p(a-b)$ when $ a \equiv b \bmod p$ and $ (ab,p)=1$.
Hence $ v_p( (p-1)^p - (-1)^p ) = v_p(p) + v_p(p-1-(-1))=2$. Hence $ p-1 \le 2$. And then $ p=3$ gives the solution $ (x,p)=(3,3)$.
Now assume that $ q \mid p-2$. Then $ q^{p-1} \mid (p-1)^{2x} - 1^{2x}$.
So $ v_q( (p-q)^{2x} - 1^{2x} ) = v_q(p-2) + v_q(2x) \ge p-1$.
But since $ q$ is odd: $ v_q(2x) = v_q(x)$. And $ q \ge 3$. Hence $ v_q(n) \le log_3(n)$ when $ n \in \mathbb{N}$. So:
$ log_3(p-2) + log_3(x) \ge p-1$.
$ x \le 2p$: $ log_3(p-2) + log_3(2p) \ge p-1 \iff 2p^2-4p \ge 3^{p-1}$
Let $ f(p) = 3^{p} - 2(p+1)^2+4(p+1) = 3^p - 2p^2+2$. Then $ f(p-1) \le 0$.
$ f'(p) = \ln (3) \cdot 3^p -4p$ and $ f''(p) = \ln^2(3) \cdot 3^p > 0$
But $ f'(2) = \ln (3) \cdot 3^2 - 4 \cdot 2 > 3^2 - 4 \cdot 2 = 1 > 0$. Since $ f''(p) > 0$ we see that $ f(p)$ is increasing when $ p \ge 2$. But $ f(2) = 3^2-2\cdot2^2+2 = 3 > 0$
Hence: When $ p \ge 3$ we see that $ f(p) > 0$. So $ q \mid p-2$ doesn't give any solutions. QED

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ropro01
62 posts
#7 • 3 Y
Y by navi_09220114, Adventure10, cubres
Setting $n=1$ we get the solutions $(1,p)$ with $p$ prime. For $p\in\{2,3\}$ we find that $(2,2)$ and $(3,3)$ are solutions too. Now let $n>1$ and $p>3$, then $n$ has to be odd.

Case 1. $n$ is a prime power, so write $n=q^\alpha$. Then we get $(p-1)^{q^\alpha}\equiv -1 \mod q \Leftrightarrow p\equiv 0 \mod q \Leftrightarrow p=q$, implying that $\alpha=1$. Now $(p-1)^p+1=\sum_{i=0}^p \binom{p}{i} p^i (-1)^{p-i}+1\equiv \sum_{i=3}^{p-2} \binom{p}{i}p^i (-1)^{p-i} - \frac{p-1}{2} p^3+p^2 \equiv 0 \mod p^{p-1}$
and as the LHS is divisible by $p^2$ but not by $p^3$, the congruence cannot hold, so there are no solutions.

Case 2. $n$ has at least two distinct prime factors. Let $q$ be the smallest prime divisor of $n$, then $n=a\cdot q^\alpha$ with $\gcd(q,a)=1$ and $a\ge 3$, so $3q^\alpha<2p$. Now we have $(p-1)^{aq^\alpha}\equiv -1 \mod q^{\alpha(p-1)}$. Squaring, we get $(p-1)^{2aq^\alpha}\equiv 1 \mod q^{\alpha(p-1)}$. As $ord_{q^{\alpha(p-1)}}(p-1)|\gcd\left(2aq^\alpha,\varphi\left(q^{\alpha(p-1)}\right)\right)=\gcd\left(2aq^\alpha,(q-1)q^{\alpha(p-1)-1}\right)=2q^\alpha$ we also have $(p-1)^{2q^\alpha}\equiv 1 \mod q^{\alpha(p-1)}\Leftrightarrow\left((p-1)^{q^\alpha}-1\right)\left((p-1)^{q^\alpha}+1\right)\equiv 0 \mod q^{\alpha(p-1)}$.
As $gcd\left((p-1)^{q^\alpha}-1,(p-1)^{q^\alpha}+1\right)=1$ we get $(p-1)^{q^\alpha}\equiv \pm 1 \mod q^{\alpha(p-1)}$. If $(p-1)^{q^\alpha}\equiv 1 \mod q^{\alpha(p-1)}$ it follows that $(p-1)^{aq^\alpha}\equiv 1 \equiv -1 \mod q^{\alpha(p-1)}$, a contradiction. But now $q^{\alpha(p-1)}|(p-1)^{q^\alpha}+1$. We know from Catalan's conjecture that equality cannot hold here, further $(q^\alpha)^{p-1}>(p-1)^{q^\alpha}\Leftrightarrow\frac{\ln(q^\alpha)}{q^\alpha}>\frac{\ln(p-1)}{p-1}$ which is clearly true, as $f'(x)= \left(\frac{\ln(x)}{x}\right)'=\frac{1-\ln(x)}{x^2}$, so $f(x)$ is strictly decreasing for $x>e$. We conclude that there are no more solutions.

Hence, $(1,p),(2,2),(3,3)$ are the only solutions. $\square$
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trigonometry456103
349 posts
#8 • 3 Y
Y by Adventure10, Mango247, cubres
Here is my solution(not exactly written up in the most organized way):
Click to reveal hidden text
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frill
316 posts
#9 • 5 Y
Y by Minkowsi47, Adventure10, Mango247, endless_abyss, cubres
Once the cases $x=1$ and $p\mid x$ have been dealt with using, for example, LTE (giving solutions of $\left(1,p\right),\left(2,2\right)$ and $\left(3,3\right)$) and it has been noted that $x$ is otherwise odd, another way to finish it off is to use orders:

$x^{p-1}\mid\left(p-1\right)^{x}+1\mid\left(p-1\right)^{2x}-1$. Thus if $q$ is the smallest a prime divisor of $x$ (the case $x=1$ has already been dealt with) then we have that $q\mid\left(p-1\right)^{x}+1\Rightarrow q\nmid\left(p-1\right)^{x}-1$ ($x$ is odd means $q\neq2$) and $q\mid\left(p-1\right)^{2x}-1$. Thus $\text{ord}_{q}\left(p-1\right)\nmid x$ and $\text{ord}_{q}\left(p-1\right)\mid2x$ hence, as $x$ is odd, $\text{ord}_{q}\left(p-1\right)$ is twice a factor of $x$ so $\frac{\text{ord}_{q}\left(p-1\right)}{2}$ is a factor of $x$ so is either $1$ or at least $q$.

Now by FLT: $\frac{\text{ord}_{q}\left(p-1\right)}{2}\mid\text{ord}_{q}\left(p-1\right)\mid q-1$ so $\frac{\text{ord}_{q}\left(p-1\right)}{2}\leq q-1\Rightarrow\frac{\text{ord}_{q}\left(p-1\right)}{2}=1\Rightarrow\text{ord}_{q}\left(p-1\right)=2$.

Now $\text{ord}_{q}\left(p-1\right)=2\Rightarrow\left(p-1\right)^{2}\equiv1\mod{q}\Rightarrow\left(p-1\right)^{x-1}\equiv1\mod{q}$ ($x$ is odd means $x-1$ is even). Thus $\left(p-1\right)^{x}\equiv p-1\mod{q}\Rightarrow\left(p-1\right)^{x}+1\equiv p\mod{q}$. But $q\mid\left(p-1\right)^{x}+1$ so $q\mid p\Rightarrow q=p\Rightarrow p\mid x$ which is a case that has already been dealt with.
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sayantanchakraborty
505 posts
#10 • 3 Y
Y by Tellocan, Adventure10, cubres
frill,your solution is really nice.....I like it.... BTW I don,t know why no one thought of using orders earlier....
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Blitzkrieg97
236 posts
#11 • 2 Y
Y by Adventure10, cubres
Davron wrote:
@all : can you please explain :-)

$q|[(p-1)^2x-1,(p-1)^{q-1}-1)=(p-1)^2-1$

Sincerely Davron
$q|(p-1)^{2x}-1$ because $(p-1)^{2x}-1=((p-1)^x+1)((p-1)^x-1)$ and $q|(p-1)^{q-1}-1$ from fermat's little theorem,so $q$ will divide GCD of them
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Dukejukem
695 posts
#12 • 3 Y
Y by Adventure10, Mango247, cubres
First, note that if $x = 1$, then the desired relation trivially holds for all $p.$ Hence, the pair $(1, p)$ is valid for all primes $p.$ Now, let us consider the cases of $x = p$ and $p = 2, 3.$ If $x = p \ne 2, 3$, then we must have $p^{p - 1} \mid (p - 1)^p + 1.$ But note that $p \mid (p - 1) + 1$, so we may apply LTE to find that $v_p\left((p - 1)^p + 1^p\right) = v_p(p) + v_p(p) = 2.$ It follows that $p - 1 \ge 2$, which is impossible from the assumption that $p \ne 2, 3.$

If $p = 3$, we are searching for $x \in \{1, 2, 3, 4, 5, 6\}$ such that $x^2 \mid 2^x + 1.$ By testing these six possibilities, we find the solutions $(1, 3), (3, 3).$ If $p = 2$, we are searching for $x \in \{1, 2, 3, 4\}$ such that $x \mid 2.$ Thus we obtain the solutions $(1, 2), (2, 2).$ Furthermore, note that if $x$ is even, then $x^{p - 1} \mid (p - 1)^x + 1 \implies p - 1$ is odd, and so $p = 2$, which is a case that was already covered. Henceforward, we may assume that $x$ is odd.

Now, suppose that $x > 1$, and let $q \ne 2, p$ be the least prime divisor of $x.$ Then we have the relation \[(p - 1)^x \equiv -1 \pmod{q^{p - 1}} \implies (p - 1)^x \equiv -1 \pmod{q} \implies (p - 1)^{2x} \equiv 1 \pmod{q}.\] Hence, if we let $d = \text{ord}_q(2)$, it is well-known that $d \mid 2x.$ Furthermore, since $p - 1$ and $q$ are clearly relatively prime, we derive that \[(p - 1)^{q - 1} \equiv 1 \pmod{q} \implies d \mid q - 1 \implies d \le q - 1.\] Now, note that if $d \ne 2$, then there is some prime divisor of $d$ that is also a divisor of $x.$ However, this is impossible, because said divisor would have to be less than $q$ (because $d \le q - 1$), and we assumed that $q$ was the least prime divisor of $x.$ Therefore, we conclude that $d = 2.$ It follows that \[(p - 1)^2 \equiv 1 \pmod{q} \implies p(p - 2) \equiv 0 \pmod{q} \implies p - 2 \equiv 0 \pmod{q},\] because we assumed that $q \ne p.$ It follows that $(p - 1)^x + 1 \equiv 1^x + 1 \equiv 2 \pmod{q}.$ However, this contradicts $(p - 1)^x + 1 \equiv 0 \pmod{q}$, so we conclude that there are no such solutions.

In summary, the desired pairs are $(2, 2), (3, 3)$, and $(1, p)$ for any prime $p.$
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Vfire
1354 posts
#13 • 3 Y
Y by AIME12345, Adventure10, cubres
1999 ISL NT 1 wrote:
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

If $x=p$ then we have $p^{p-1} \mid (p-1)^p+1$. Since $p \mid (p-1)+1$, by LTE we know that $v_p((p-1)^p+1) = v_p((p-1)+1)+v_p(p) = 2$. Therefore, $p-1 \le 2 \implies p=2,3$. These yield the valid pairs $\boxed{(2,2), (3,3)}$. If $x=1$ then all pairs in the form $\boxed{(1,p)}$ where $p$ is a prime work. Now assume $p >2, x>1$ and $p \neq x$.

We have $x^{p-1} \mid (p-1)^x+1 \mid (p-1)^{2x}-1$. Notice that $x$ must be odd because $(p-1)^x+1$ is odd ($p >2$). Therefore, $\gcd(x^{p-1}, (p-1)^x-1)=1$ because $x^{p-1} \mid (p-1)^x+1$ and the smallest prime that divides $x^{p-1}$ is greater than $2$. Then if $q$ is the smallest prime factor of $x$, we know that $\text{ord}_q(p-1) \mid 2x$ and $\text{ord}_q(p-1) \nmid x$. Since $x$ is odd, we know that $\text{ord}_q(p-1) \ge 2q$ or $\text{ord}_q(p-1)=2$. However, since $\phi(q) = q-1$, we know that $\text{ord}_q(p-1) \mid q-1$ so we must have $\text{ord}_q(p-1)=2$. Hence, $(p-1)^2 \equiv 1 \pmod{q}$. Then $(p-1)^x+1 \equiv 0 \pmod{q} \implies p \equiv 0 \pmod{q}$. Since $p$ is prime, $p=q$ so $p$ must also equal $x$ since $x \le 2p$, which goes against our assumptions. Therefore, we are done. $\blacksquare$
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samoha
9 posts
#15 • 3 Y
Y by Adventure10, Mango247, cubres
Hey, noob here. A bit off-topic question, but is there something wrong with this statement below? (Not that it's useful)

This is for cases $x > 1, p > 2$.
$x^{p - 1} \mid \left(p - 1\right)^x + 1 \Rightarrow \nu_p(x^{p - 1}) \leq \nu_p(\left(p - 1\right)^x + 1^x) = \nu_p(p) +\nu_p(x) = 1 + \nu_p(x) \Rightarrow \left(p - 2 \right)\nu_p(x) \leq 1$
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math_pi_rate
1218 posts
#16 • 3 Y
Y by Adventure10, Mango247, cubres
My solution: If $p=2$, then $x \mid 2 \Rightarrow x=1,2$. So from now on assume that $p$ is odd.

Now, $x^{p - 1} \mid \left(p - 1\right)^x + 1 \Rightarrow (p-1) \cdot \nu_p(x) = \nu_p(x^{p - 1}) \leq \nu_p(\left(p - 1\right)^x + 1) = \nu_p(p) +\nu_p(x) = 1 + \nu_p(x) \Rightarrow \left(p - 2 \right)\nu_p(x) \leq 1$.

CASE-1 ($\nu_p(x) \geq 1$): $p-2 \leq 1 \Rightarrow p \leq 3 \Rightarrow p=3$.

And, $p \mid x \Rightarrow 3 \mid x$ and $x \leq 2p = 6$ $\Rightarrow x=3$ or $6$. But $x^2 \mid 2^x+1 \Rightarrow x=3$

CASE-2 ($\nu_p(x)=0$): Notice that $x=1$ works for all $p$, so from now on assume that $x \not= 1$.

Let $m$ be a prime divisor of $x$. Note that as $p \nmid x$, so $p \not= m$. Then, $(p-1)^x+1 \equiv 0 \pmod{m} \Rightarrow (p-1)^{2x} \equiv 1 \pmod{m}$

Thus, $ord_m(p-1) \mid 2x$. Also $ord_m(p-1) \mid \phi(m) \Rightarrow ord_m(p-1) \mid m-1$.

Also as $ord_m(p-1) \leq m-1$, so $ord_m(p-1) \mid gcd(m-1,2x)$. But as $m \mid x \Rightarrow m-1 \nmid x$.

SUBCASE-1 ($m \not= 3$): $m-1 \nmid 2 \Rightarrow gcd(m-1,2x)=1 \Rightarrow ord_m(p-1) = 1 \Rightarrow p \equiv 1 \pmod{m}$

$\Rightarrow (p-1)^x+1 \equiv 1 \equiv 0 \pmod{m} \longrightarrow CONTRADICTION$.

SUBCASE-2 ($m=3$): $m-1 \mid 2 \Rightarrow gcd(m-1,2x) =2 \Rightarrow ord_m(p-1)=1 \text{or} 2$.

By a similar argument as in subcase-1, $ord_m(p-1) \not= 1 \Rightarrow ord_m(p-1) = 2 \Rightarrow (p-1)^2 \equiv 1 \pmod{m}$

$\Rightarrow p(p-2) \equiv 0 \pmod{m} \Rightarrow p-2 \equiv 0 \pmod{m} \Rightarrow (p-1)^x+1 \equiv 2 \equiv 0 \pmod{m} \longrightarrow CONTRADICTION$.

Thus, The only solutions are $(x,p)=(1,p),(2,2),(3,3)$.
This post has been edited 1 time. Last edited by math_pi_rate, Aug 23, 2018, 3:57 PM
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