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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
Cyclic Quads and Parallel Lines
gracemoon124   16
N 7 minutes ago by ohiorizzler1434
Source: 2015 British Mathematical Olympiad?
Let $ABCD$ be a cyclic quadrilateral. Let $F$ be the midpoint of the arc $AB$ of its circumcircle which does not contain $C$ or $D$. Let the lines $DF$ and $AC$ meet at $P$ and the lines $CF$ and $BD$ meet at $Q$. Prove that the lines $PQ$ and $AB$ are parallel.
16 replies
1 viewing
gracemoon124
Aug 16, 2023
ohiorizzler1434
7 minutes ago
Radical Center on the Euler Line (USEMO 2020/3)
franzliszt   37
N 27 minutes ago by Ilikeminecraft
Source: USEMO 2020/3
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly.
Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$.

Proposed by Anant Mudgal
37 replies
franzliszt
Oct 24, 2020
Ilikeminecraft
27 minutes ago
Functional equation with powers
tapir1729   13
N 28 minutes ago by ihategeo_1969
Source: TSTST 2024, problem 6
Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$,
\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]Jaedon Whyte
13 replies
tapir1729
Jun 24, 2024
ihategeo_1969
28 minutes ago
Powers of a Prime
numbertheorist17   34
N an hour ago by KevinYang2.71
Source: USA TSTST 2014, Problem 6
Suppose we have distinct positive integers $a, b, c, d$, and an odd prime $p$ not dividing any of them, and an integer $M$ such that if one considers the infinite sequence \begin{align*}
		ca &- db \\
		ca^2 &- db^2 \\
		ca^3 &- db^3 \\
		ca^4 &- db^4 \\
&\vdots
	\end{align*} and looks at the highest power of $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a,b,c,d,p,M$) such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^T$.
34 replies
+1 w
numbertheorist17
Jul 16, 2014
KevinYang2.71
an hour ago
q(x) to be the product of all primes less than p(x)
orl   18
N an hour ago by happypi31415
Source: IMO Shortlist 1995, S3
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
18 replies
orl
Aug 10, 2008
happypi31415
an hour ago
IMO 2018 Problem 5
orthocentre   80
N an hour ago by OronSH
Source: IMO 2018
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

Proposed by Bayarmagnai Gombodorj, Mongolia
80 replies
orthocentre
Jul 10, 2018
OronSH
an hour ago
Line passes through fixed point, as point varies
Jalil_Huseynov   60
N 2 hours ago by Rayvhs
Source: APMO 2022 P2
Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.
60 replies
Jalil_Huseynov
May 17, 2022
Rayvhs
2 hours ago
Tangent to two circles
Mamadi   2
N 2 hours ago by A22-
Source: Own
Two circles \( w_1 \) and \( w_2 \) intersect each other at \( M \) and \( N \). The common tangent to two circles nearer to \( M \) touch \( w_1 \) and \( w_2 \) at \( A \) and \( B \) respectively. Let \( C \) and \( D \) be the reflection of \( A \) and \( B \) respectively with respect to \( M \). The circumcircle of the triangle \( DCM \) intersect circles \( w_1 \) and \( w_2 \) respectively at points \( E \) and \( F \) (both distinct from \( M \)). Show that the line \( EF \) is the second tangent to \( w_1 \) and \( w_2 \).
2 replies
Mamadi
May 2, 2025
A22-
2 hours ago
Deduction card battle
anantmudgal09   55
N 3 hours ago by deduck
Source: INMO 2021 Problem 4
A Magician and a Detective play a game. The Magician lays down cards numbered from $1$ to $52$ face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.

Prove that the Detective can guarantee a win if and only if she is allowed to ask at least $50$ questions.

Proposed by Anant Mudgal
55 replies
1 viewing
anantmudgal09
Mar 7, 2021
deduck
3 hours ago
Geometry
Lukariman   7
N 3 hours ago by vanstraelen
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 $\angle HDM$ = 2∠AMP.
7 replies
Lukariman
Yesterday at 12:43 PM
vanstraelen
3 hours ago
perpendicularity involving ex and incenter
Erken   20
N 4 hours ago by Baimukh
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
20 replies
Erken
Dec 24, 2008
Baimukh
4 hours ago
Isosceles Triangle Geo
oVlad   4
N 4 hours ago by Double07
Source: Romania Junior TST 2025 Day 1 P2
Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$
4 replies
oVlad
Apr 12, 2025
Double07
4 hours ago
Geometry
Lukariman   1
N 4 hours ago by Primeniyazidayi
Given acute triangle ABC ,AB=b,AC=c . M is a variable point on side AB. The circle circumscribing triangle BCM intersects AC at N.

a)Let I be the center of the circle circumscribing triangle AMN. Prove that I always lies on a fixed line.

b)Let J be the center of the circle circumscribing triangle MBC. Prove that line segment IJ has a constant length.
1 reply
Lukariman
Today at 4:02 PM
Primeniyazidayi
4 hours ago
Kingdom of Anisotropy
v_Enhance   24
N 4 hours ago by deduck
Source: IMO Shortlist 2021 C4
The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from $X$ to $Y$ is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.

Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.

Proposed by Warut Suksompong, Thailand
24 replies
v_Enhance
Jul 12, 2022
deduck
4 hours ago
gcd (a^n+b,b^n+a) is constant
EthanWYX2009   81
N May 4, 2025 by GingerMan
Source: 2024 IMO P2
Determine all pairs $(a,b)$ of positive integers for which there exist positive integers $g$ and $N$ such that
$$\gcd (a^n+b,b^n+a)=g$$holds for all integers $n\geqslant N.$ (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y.$)

Proposed by Valentio Iverson, Indonesia
81 replies
EthanWYX2009
Jul 16, 2024
GingerMan
May 4, 2025
gcd (a^n+b,b^n+a) is constant
G H J
G H BBookmark kLocked kLocked NReply
Source: 2024 IMO P2
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straight
414 posts
#76 • 1 Y
Y by cubres
Assassino9931 wrote:
Did anyone mention that this problem appeared in the IMSC China International camp 2024 and benefitted some students??? :)

(P.S. I am not blaming anyone that a setup has been made; in fact I can assure that no such thing has happened, I am just giving a fun fact here.)

@above I also realized this and I think it's very unfair indeed. It is completely the same idea with completely the same construction. If I recall proposer of that problem was Navid Safaei (forgive me if I'm wrong) . So far I haven't heard of anyone claiming to have known this problem before the IMO though
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Assassino9931
1324 posts
#77 • 1 Y
Y by cubres
straight wrote:
Assassino9931 wrote:
Did anyone mention that this problem appeared in the IMSC China International camp 2024 and benefitted some students??? :)

(P.S. I am not blaming anyone that a setup has been made; in fact I can assure that no such thing has happened, I am just giving a fun fact here.)

@above I also realized this and I think it's very unfair indeed. It is completely the same idea with completely the same construction. If I recall proposer of that problem was Navid Safaei (forgive me if I'm wrong) . So far I haven't heard of anyone claiming to have known this problem before the IMO though

No, the origin of the problem I put is Ukraine 2019 8.8/9.7 by Arsenii Nikolaiev (who was actually Observer B at IMO 2024 if I am not mistaken, so in particular not part of the leaders/observer A problem voting). And, well, I did hear about at least one student benefitting from that, but prefer keep the identity of the country confidential.
This post has been edited 1 time. Last edited by Assassino9931, Mar 31, 2025, 10:53 AM
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Cali.Math
128 posts
#78 • 1 Y
Y by cubres
We uploaded our solution https://calimath.org/pdf/IMO2024-2.pdf on youtube https://youtu.be/daboPS8Dtyk.
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Assassino9931
1324 posts
#79 • 3 Y
Y by pavel kozlov, VicKmath7, cubres
Here is the (primitive root)-styled solution, in spirit of what VicKmath7 wrote above.

Let $p \geq 3$ be a prime. Consider firstly the following: when is there an integer $n$ such that $a^n + b \equiv 0 \pmod p$? If $g$ is a primitive root mod $p$ (where $p$ is a prime not dividing $a$ or $b$!), then writing $a \equiv g^A$, $b \equiv g^B$ transfers to the equivalent $g^{B}(g^{nA-B}+1) \equiv 0 \pmod p$, i.e. $nA - B \equiv \frac{p-1}{2} \pmod {p-1}$. Similarly, for $b^n+a$ to be divisible by $p$ we must have $nB - A \equiv \frac{p-1}{2} \pmod {p-1}$. Now if it is the case that $A+B\equiv \frac{p-1}{2} \pmod {p-1}$, then taking $n\equiv -1 \pmod {p-1}$ would work not only for $a^n+b$, but also for $b^n + a$.

But note that $A + B \equiv \frac{p-1}{2} \pmod {p-1}$ if and only if $g^{A+B} \equiv -1 \pmod p$, i.e. $p$ divides $ab+1$. Therefore if we initially take $p$ to be an odd prime divisor of $ab+1$ (note that such does not divide $a$ or $b$), then there are infinitely many $n$, for which the required greatest common divisor is divisible by $p$. However, it cannot be the case that $p$ divides $a^n+b$ and $b^n+a$ for all large $n$ -- otherwise, $p$ would divide $a^{n+1} + ab \equiv a^{n+1} - 1$, so $p$ would divide $a-1$ (due to $a^{n+1} \equiv a^{n+2} \equiv 1 \pmod p$ and $\gcd(a,ab+1) = 1$), similarly $p$ would divide $b-1$, but now $ab+1 \equiv 0 \pmod p$ implies that $p$ must also divide $a+1$ and hence $(a+1) - (a-1) = 2$, contradicting $p\geq 3$.

Therefore, if $a$ and $b$ are such that $ab+1$ has an odd prime divisor, then they cannot satisfy the problem conditions. Finally, suppose $ab+1$ is a power of $2$. Fortunately, we only need that $ab+1$ is divisible by $4$ (unless $a=b=1$, which satisfies the problem conditions). Indeed, we may assume $a\equiv 3 \pmod 4$ and $b \equiv 1 \pmod 4$ and now if $n$ is even, then $a^n + b$ is divisible by $2$, but not by $4$ (and $b^n + a$ is divisible by $4$), while if $n$ is odd, then both $a^n + b$ and $b^n + a$ are divisible by $4$, so $\gcd(a^n+b,b^n+a)$ is infinitely often divisible by $4$ and not divisible by $4$, contradiction.
This post has been edited 3 times. Last edited by Assassino9931, Sep 18, 2024, 8:22 AM
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L13832
268 posts
#80 • 2 Y
Y by radian_51, cubres
What an amazing problem, kudos to the problem proposer :)
$\textbf{Answer:}$ $a=b=1$
Let $p$ be a prime such that $p\mid g$, so $p\mid b^{n-1}(a^n+b)-b^n-a\implies p\mid ab-1$ if $p\nmid a$.
Finally checking if $ab+1\mid g$ and choosing $n\equiv -1\pmod{\phi(ab+1)}$(this is possible because $\gcd(ab+1,a)=\gcd(b,ab+1)=1$) we have
$$a^n+b\equiv a^{-1}+b\equiv \frac{ab+1}{a}\equiv 0\pmod{ab+1}$$$$b^n+1\equiv a+b^{-1}\equiv \frac{ab+1}{b}\equiv 0\pmod{ab+1}$$Motivation
The last part comes from the fact that $a,b$ can be inverted modulo ${ab+1}$. Similarly we obtain $b^n+a\equiv 0\pmod{ab+1}$ which gives us $ab+1\mid g\implies p\mid ab+1$
Now we take $n \equiv 0 \pmod{p-1}$, we get that $p \mid g  = \gcd(a^n + b, b^n + a)$, so by FLT, $p \mid a + 1, p \mid b + 1 \implies p \mid 2\implies p=2$.
If $ab+1=2$ then $\boxed{a=b=1}$
Otherwise $ab+1\equiv 0\pmod{4}\implies a,b\equiv \pm 1\pmod{4}$.
WLOG $a \equiv -1 \pmod{4}$ and $b \equiv 1 \pmod{4}$. If $n$ is odd then we obtain that $a^n + b$ and $b^n + a$ are both divisible by $4$, and therefore $4 \mid g$. But having $n$ even we get $a^n + b \equiv 2 \pmod{4}$ and $4 \nmid a^n + b$, a contradiction, so we are done! :yoda:
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Golden_Verse
5 posts
#81 • 1 Y
Y by cubres
Answer
Solution
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Vedoral
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#82 • 1 Y
Y by cubres
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AshAuktober
1005 posts
#83 • 1 Y
Y by cubres
Note that $ab+1 \mid g \mid a-b$, so indeed we must have $a  =b $, from where the only working pair is $(1, 1).
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zaidova
87 posts
#84 • 1 Y
Y by cubres
For all n, which are positive integers ($a=b=1$) is the only solution.
This post has been edited 3 times. Last edited by zaidova, Jan 3, 2025, 7:43 PM
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cursed_tangent1434
623 posts
#85 • 1 Y
Y by ihategeo_1969
I just can't believe I didn't manage to solve this problem in contest. The idea of considering $ab+1$ is not that random either, especially since it is almost natural to consider $a+b$ at which point you decide that the considered expression had better be symmetric but also relatively prime to both $a$ and $b$, which leads to the considered form.

The entirety of the problem is the following key claim.

Claim : For any such pair $(a,b)$ we must have $ab+1$ a power of two.

Proof : Consider a prime $p\mid ab+1$. Then, since $\gcd(ab+1,a)=\gcd(ab+1,b)=1$ it follows that $p\nmid a,b$ and so letting $n=k(p-1)-1$ for sufficiently large positive integers $k$ we have,
\[a^n+b \equiv \frac{1}{a}+b \equiv \frac{ab+1}{a} \equiv 0 \pmod{p}\]and
\[a+b^n \equiv a+\frac{1}{b} \equiv \frac{ab+1}{b} \equiv 0 \pmod{p}\]which implies that $p \mid \gcd(a^n+b,b^n+a)$. But then, if the $\gcd$ is eventually constant, we have that $p \mid \gcd(a^n+b,b^n+a)$ for all sufficiently large positive integers $n$. But then, considering $n=k(p-1)$ for sufficiently large positive integers $k$ we have that,
\[0 \equiv a^n+b \equiv b+1 \pmod{p}\]And similarly, $a+1 \equiv 0 \pmod{p}$. But then, since $p\mid ab+1$ and $p \mid a+1$ it follows that $p \mid b-1$ which in conjunction with $p\mid b+2$ implies that $p=2$. This means that there cannot exist any odd prime $p$ dividing $ab+1$ proving the claim.

Now, if $ab+1>2$ and
\[ab+1=2^r\]for some $r \ge 2$, it follows that $ab \equiv 3 \pmod{4}$. Thus, $a\equiv 1 \pmod{4}$ and $b \equiv 3 \pmod{4}$ (or vice versa). But note that this means for even $n$,
\[a^n + b \equiv 1+3 \equiv 0 \pmod{4}\]but
\[a+b^n \equiv 1+1 \equiv 2 \pmod{4} \]Thus, $\nu_2(\gcd(a^n+b,b^n+a))=1$. But note that for odd $n$ we have,
\[a^n+b \equiv 1 + 3 \equiv 0 \pmod{4}\]and
\[a+b^n \equiv 1 + 3 \equiv 0 \pmod{4}\]which implies that $4\mid \gcd(a^n+b,b^n+a)$. But, if the $\gcd$ is eventually constant this is a clear contradiction, which implies that we must have $ab+1=2$ and thus, $a=b=1$ as desired.
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quantam13
112 posts
#86
Y by
Cute solution. The main idea is to consider the sequence modulo $ab+1$, which can be motivated by "$n=-1$".
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dstanz5
243 posts
#89
Y by
quantam13 wrote:
Cute solution. The main idea is to consider the sequence modulo $ab+1$, which can be motivated by "$n=-1$".

Sorry for necroposting, I am going through the recent problems and I wish to know more about why $ab+1$ is the key expression here. "$n = -1$" would give $gcd(\frac{1}{a} + b, \frac{1}{b} + a)$ but I don't know what this gives.
EDIT: Oh I see, if you multiply them by a and b respectively they both give $ab+1$. Thanks to khina and vEnhance
This post has been edited 1 time. Last edited by dstanz5, Mar 31, 2025, 7:51 AM
Reason: :(
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Ilikeminecraft
617 posts
#90
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I claim that $(a, b) = (1, 1)$ is the only valid answer.

We take $((a^c + b, b^c + a), (a^{c + 1} + b, b^{c + 1} + a))$ first. Note that $((a, b), (c, d)) = ((a, c), (b, d)).$ Thus,
\begin{align*}
  ((a^c + b, b^c + a), (a^{c + 1} + b, b^{c + 1} + a)) & = ((a^c + b, a^{c + 1} + b), (b^c + a, b^{c + 1} + a)) \\
  (a^c + b, a^{c + 1} + b) & = (a^c + b, a^{c + 1} - a^c) \\
  & = (a^c + b, a - 1)(a^c + b, a^c) \text{ since } (a^c, a - 1) = 1\\
  & = (a^{c - 1} + b, a - 1)(b, a^c) \\
  & = (b + 1, a - 1)(b, a^c) \\
  ((a^c + b, b^c + a), (a^{c + 1} + b, b^{c + 1} + a)) & = ((b + 1, a - 1)(b, a^c), (b - 1, a + 1)(a, b^c)) \\
  & \mid ((b + 1, a-1),(b - 1, a + 1)(a, b^c))((b, a^c),(b - 1, a + 1)(a, b^c)) \\
  & \mid ((b + 1, a-1),(b - 1, a + 1))((b, a^c), (a, b^c)) \\
  & = 2(a, b)
\end{align*}Thus, we have that $(a^n + b, b^n + a) \mid 2(a, b)$ when it becomes constant. Now, let $q = ab + 1.$ By picking an $n \equiv-1\pmod {\phi(q)},$ we have that $q\mid a + \frac1b, b + \frac1a \implies q\mid(a^n + b, b^n + a) \implies q \mid 2(a, b).$ However, we have that $q = ab + 1\implies (q, a) = (q, b) = 1.$ Thus, either $q = 1,$ or 2. If $q = 1,$ then $a$ or $b$ aren't positive. Thus, $q = 2\implies \boxed{a = b = 1}.$
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santhoshn
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#91
Y by
1, 1 Answer
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GingerMan
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#92
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Answer is $(a,b)=(1,1)$ only, which works.
Take any prime divisor $p$ of $ab+1$ and let $t=\nu_p(ab+1)$. Then
\begin{align*}
  a^n+b &\equiv a^n - \frac 1a \equiv \frac{a^{n+1}-1}{a} \pmod{p^t}\\
  b^n+a &\equiv b^n - \frac 1b \equiv \frac{b^{n+1}-1}{b} \pmod{p^t}
\end{align*}So $p^t\mid \gcd(a^n+b,b^n+a)$ if and only if $\operatorname{lcm}(x,y) \mid n+1$, where $x$, $y$ are the orders of $a$, $b$ mod $p^t$, respectively. If $p^t>2$ and $(a,b)\neq (1,1)$, then $\operatorname{lcm}(x,y)>1$, so the sequence is not eventually constant.
Thus $p^t=2$, implying $ab+1=2$, which still gives $(a,b)=(1,1)$.
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