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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
May 1, 2025
0 replies
k i Peer-to-Peer Programs Forum
jwelsh   157
N Dec 11, 2023 by cw357
Many of our AoPS Community members share their knowledge with their peers in a variety of ways, ranging from creating mock contests to creating real contests to writing handouts to hosting sessions as part of our partnership with schoolhouse.world.

To facilitate students in these efforts, we have created a new Peer-to-Peer Programs forum. With the creation of this forum, we are starting a new process for those of you who want to advertise your efforts. These advertisements and ensuing discussions have been cluttering up some of the forums that were meant for other purposes, so we’re gathering these topics in one place. This also allows students to find new peer-to-peer learning opportunities without having to poke around all the other forums.

To announce your program, or to invite others to work with you on it, here’s what to do:

1) Post a new topic in the Peer-to-Peer Programs forum. This will be the discussion thread for your program.

2) Post a single brief post in this thread that links the discussion thread of your program in the Peer-to-Peer Programs forum.

Please note that we’ll move or delete any future advertisement posts that are outside the Peer-to-Peer Programs forum, as well as any posts in this topic that are not brief announcements of new opportunities. In particular, this topic should not be used to discuss specific programs; those discussions should occur in topics in the Peer-to-Peer Programs forum.

Your post in this thread should have what you're sharing (class, session, tutoring, handout, math or coding game/other program) and a link to the thread in the Peer-to-Peer Programs forum, which should have more information (like where to find what you're sharing).
157 replies
jwelsh
Mar 15, 2021
cw357
Dec 11, 2023
k i C&P posting recs by mods
v_Enhance   0
Jun 12, 2020
The purpose of this post is to lay out a few suggestions about what kind of posts work well for the C&P forum. Except in a few cases these are mostly meant to be "suggestions based on historical trends" rather than firm hard rules; we may eventually replace this with an actual list of firm rules but that requires admin approval :) That said, if you post something in the "discouraged" category, you should not be totally surprised if it gets locked; they are discouraged exactly because past experience shows they tend to go badly.
-----------------------------
1. Program discussion: Allowed
If you have questions about specific camps or programs (e.g. which classes are good at X camp?), these questions fit well here. Many camps/programs have specific sub-forums too but we understand a lot of them are not active.
-----------------------------
2. Results discussion: Allowed
You can make threads about e.g. how you did on contests (including AMC), though on AMC day when there is a lot of discussion. Moderators and administrators may do a lot of thread-merging / forum-wrangling to keep things in one place.
-----------------------------
3. Reposting solutions or questions to past AMC/AIME/USAMO problems: Allowed
This forum contains a post for nearly every problem from AMC8, AMC10, AMC12, AIME, USAJMO, USAMO (and these links give you an index of all these posts). It is always permitted to post a full solution to any problem in its own thread (linked above), regardless of how old the problem is, and even if this solution is similar to one that has already been posted. We encourage this type of posting because it is helpful for the user to explain their solution in full to an audience, and for future users who want to see multiple approaches to a problem or even just the frequency distribution of common approaches. We do ask for some explanation; if you just post "the answer is (B); ez" then you are not adding anything useful.

You are also encouraged to post questions about a specific problem in the specific thread for that problem, or about previous user's solutions. It's almost always better to use the existing thread than to start a new one, to keep all the discussion in one place easily searchable for future visitors.
-----------------------------
4. Advice posts: Allowed, but read below first
You can use this forum to ask for advice about how to prepare for math competitions in general. But you should be aware that this question has been asked many many times. Before making a post, you are encouraged to look at the following:
[list]
[*] Stop looking for the right training: A generic post about advice that keeps getting stickied :)
[*] There is an enormous list of links on the Wiki of books / problems / etc for all levels.
[/list]
When you do post, we really encourage you to be as specific as possible in your question. Tell us about your background, what you've tried already, etc.

Actually, the absolute best way to get a helpful response is to take a few examples of problems that you tried to solve but couldn't, and explain what you tried on them / why you couldn't solve them. Here is a great example of a specific question.
-----------------------------
5. Publicity: use P2P forum instead
See https://artofproblemsolving.com/community/c5h2489297_peertopeer_programs_forum.
Some exceptions have been allowed in the past, but these require approval from administrators. (I am not totally sure what the criteria is. I am not an administrator.)
-----------------------------
6. Mock contests: use Mock Contests forum instead
Mock contests should be posted in the dedicated forum instead:
https://artofproblemsolving.com/community/c594864_aops_mock_contests
-----------------------------
7. AMC procedural questions: suggest to contact the AMC HQ instead
If you have a question like "how do I submit a change of venue form for the AIME" or "why is my name not on the qualifiers list even though I have a 300 index", you would be better off calling or emailing the AMC program to ask, they are the ones who can help you :)
-----------------------------
8. Discussion of random math problems: suggest to use MSM/HSM/HSO instead
If you are discussing a specific math problem that isn't from the AMC/AIME/USAMO, it's better to post these in Middle School Math, High School Math, High School Olympiads instead.
-----------------------------
9. Politics: suggest to use Round Table instead
There are important conversations to be had about things like gender diversity in math contests, etc., for sure. However, from experience we think that C&P is historically not a good place to have these conversations, as they go off the rails very quickly. We encourage you to use the Round Table instead, where it is much more clear that all posts need to be serious.
-----------------------------
10. MAA complaints: discouraged
We don't want to pretend that the MAA is perfect or that we agree with everything they do. However, we chose to discourage this sort of behavior because in practice most of the comments we see are not useful and some are frankly offensive.
[list] [*] If you just want to blow off steam, do it on your blog instead.
[*] When you have criticism, it should be reasoned, well-thought and constructive. What we mean by this is, for example, when the AOIME was announced, there was great outrage about potential cheating. Well, do you really think that this is something the organizers didn't think about too? Simply posting that "people will cheat and steal my USAMOO qualification, the MAA are idiots!" is not helpful as it is not bringing any new information to the table.
[*] Even if you do have reasoned, well-thought, constructive criticism, we think it is actually better to email it the MAA instead, rather than post it here. Experience shows that even polite, well-meaning suggestions posted in C&P are often derailed by less mature users who insist on complaining about everything.
[/list]
-----------------------------
11. Memes and joke posts: discouraged
It's fine to make jokes or lighthearted posts every so often. But it should be done with discretion. Ideally, jokes should be done within a longer post that has other content. For example, in my response to one user's question about olympiad combinatorics, I used a silly picture of Sogiita Gunha, but it was done within a context of a much longer post where it was meant to actually make a point.

On the other hand, there are many threads which consist largely of posts whose only content is an attached meme with the word "MAA" in it. When done in excess like this, the jokes reflect poorly on the community, so we explicitly discourage them.
-----------------------------
12. Questions that no one can answer: discouraged
Examples of this: "will MIT ask for AOIME scores?", "what will the AIME 2021 cutoffs be (asked in 2020)", etc. Basically, if you ask a question on this forum, it's better if the question is something that a user can plausibly answer :)
-----------------------------
13. Blind speculation: discouraged
Along these lines, if you do see a question that you don't have an answer to, we discourage "blindly guessing" as it leads to spreading of baseless rumors. For example, if you see some user posting "why are there fewer qualifiers than usual this year?", you should not reply "the MAA must have been worried about online cheating so they took fewer people!!". Was sich überhaupt sagen lässt, lässt sich klar sagen; und wovon man nicht reden kann, darüber muss man schweigen.
-----------------------------
14. Discussion of cheating: strongly discouraged
If you have evidence or reasonable suspicion of cheating, please report this to your Competition Manager or to the AMC HQ; these forums cannot help you.
Otherwise, please avoid public discussion of cheating. That is: no discussion of methods of cheating, no speculation about how cheating affects cutoffs, and so on --- it is not helpful to anyone, and it creates a sour atmosphere. A longer explanation is given in Seriously, please stop discussing how to cheat.
-----------------------------
15. Cutoff jokes: never allowed
Whenever the cutoffs for any major contest are released, it is very obvious when they are official. In the past, this has been achieved by the numbers being posted on the official AMC website (here) or through a post from the AMCDirector account.

You must never post fake cutoffs, even as a joke. You should also refrain from posting cutoffs that you've heard of via email, etc., because it is better to wait for the obvious official announcement. A longer explanation is given in A Treatise on Cutoff Trolling.
-----------------------------
16. Meanness: never allowed
Being mean is worse than being immature and unproductive. If another user does something which you think is inappropriate, use the Report button to bring the post to moderator attention, or if you really must reply, do so in a way that is tactful and constructive rather than inflammatory.
-----------------------------

Finally, we remind you all to sit back and enjoy the problems. :D

-----------------------------
(EDIT 2024-09-13: AoPS has asked to me to add the following item.)

Advertising paid program or service: never allowed

Per the AoPS Terms of Service (rule 5h), general advertisements are not allowed.

While we do allow advertisements of official contests (at the MAA and MATHCOUNTS level) and those run by college students with at least one successful year, any and all advertisements of a paid service or program is not allowed and will be deleted.
0 replies
v_Enhance
Jun 12, 2020
0 replies
k i Stop looking for the "right" training
v_Enhance   50
N Oct 16, 2017 by blawho12
Source: Contest advice
EDIT 2019-02-01: https://blog.evanchen.cc/2019/01/31/math-contest-platitudes-v3/ is the updated version of this.

EDIT 2021-06-09: see also https://web.evanchen.cc/faq-contest.html.

Original 2013 post
50 replies
v_Enhance
Feb 15, 2013
blawho12
Oct 16, 2017
Radical Center on the Euler Line (USEMO 2020/3)
franzliszt   37
N a few seconds 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
a few seconds ago
Functional equation with powers
tapir1729   13
N a few seconds 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
a few seconds ago
Powers of a Prime
numbertheorist17   34
N 18 minutes 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
numbertheorist17
Jul 16, 2014
KevinYang2.71
18 minutes ago
q(x) to be the product of all primes less than p(x)
orl   18
N 30 minutes 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
30 minutes ago
Looking for Physics or USAPhO Tutor
physicsplease   4
N 4 hours ago by talhee
Hii I am looking for a USAPhO tutor for next year's season. I think I have tried literally everything possible to improve but I feel like I just hit a massive roadblock right now.

It would be ideal if I can find someone who have a lot of experience with physics olympiads. My goal is medal/gold in usapho next year, and I am very determined & willing to put in a lot of hours, especially more so in the summer. Please recommend anyone or dm in aops, thank you.

Have qualified usapho before (last year), took both physics c and sufficient higher math.
4 replies
physicsplease
Apr 11, 2025
talhee
4 hours ago
9 ARML Location
deduck   35
N Today at 4:57 PM by deduck
UNR -> Nevada
St Anselm -> New Hampshire
PSU -> Pennsylvania
WCU -> North Carolina


Put your USERNAME in the list ONLY IF YOU WANT TO!!!! !!!!!

I'm going to UNR if anyone wants to meetup!!! :D

Current List:
Iowa
UNR
PSU
St Anselm
WCU
35 replies
deduck
Yesterday at 4:19 PM
deduck
Today at 4:57 PM
Jane street swag package? USA(J)MO
arfekete   1
N Today at 4:36 PM by arfekete
Hey! People are starting to get their swag packages from Jane Street for qualifying for USA(J)MO, and after some initial discussion on what we got, people are getting different things. Out of curiosity, I was wondering how they decide who gets what.
Please enter the following info:

- USAMO or USAJMO
- Grade
- Score
- Award/Medal/HM
- MOP (yes or no, if yes then color)
- List of items you got in your package

I will reply with my info as an example.
1 reply
arfekete
Today at 4:34 PM
arfekete
Today at 4:36 PM
MathILy 2025 Decisions Thread
mysterynotfound   40
N Today at 4:11 PM by bjump
Discuss your decisions here!
also share any relevant details about your decisions if you want
40 replies
mysterynotfound
Apr 21, 2025
bjump
Today at 4:11 PM
Mathcounts state
happymoose666   39
N Today at 1:54 PM by Inaaya
Hi everyone,
I just have a question. I live in PA and I sadly didn't make it to nationals this year. Is PA a competitive state? I'm new into mathcounts and not sure
39 replies
happymoose666
Mar 24, 2025
Inaaya
Today at 1:54 PM
force overlay inversion vibes
v4913   63
N Today at 1:47 PM by starchan
Source: USAMO 2023/6
Let $ABC$ be a triangle with incenter $I$ and excenters $I_a$, $I_b$, and $I_c$ opposite $A$, $B$, and $C$, respectively. Let $D$ be an arbitrary point on the circumcircle of $\triangle{ABC}$ that does not lie on any of the lines $II_a$, $I_bI_c$, or $BC$. Suppose the circumcircles of $\triangle{DII_a}$ and $\triangle{DI_bI_c}$ intersect at two distinct points $D$ and $F$. If $E$ is the intersection of lines $DF$ and $BC$, prove that $\angle{BAD} = \angle{EAC}$.

Proposed by Zach Chroman
63 replies
v4913
Mar 23, 2023
starchan
Today at 1:47 PM
Perfect squares: 2011 USAJMO #1
v_Enhance   226
N Today at 12:51 PM by RachitSuckAtMath
Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.
226 replies
v_Enhance
Apr 28, 2011
RachitSuckAtMath
Today at 12:51 PM
SUMaC Residential vs. Ross
AwesomeDude10   6
N Today at 2:54 AM by boaway123
Hi! I got into the SUMaC residential i program, and I also recently got off the Ross waitlist. I was wondering if anyone had any insight on which program is
1) More useful in furthering my mathematical knowledge (which has a better curriculum)
2) Since I'm a junior, which is more useful for college apps? (I know this is a little cringe)
Thanks!
6 replies
AwesomeDude10
Yesterday at 9:46 PM
boaway123
Today at 2:54 AM
purple comet discussion
ConfidentKoala4   65
N Today at 1:26 AM by abbominable_sn0wman
when can we discuss purple comet
65 replies
ConfidentKoala4
May 2, 2025
abbominable_sn0wman
Today at 1:26 AM
HCSSiM vs other programs
MathWizardThatCanBeatYou   1
N Yesterday at 11:44 PM by lpieleanu
For anyone that has been to HCSSiM and other summer math programs like Canada/USA Mathcamp, Ross, PROMYS, etc. What's the difference and which would you consider more worth it?
1 reply
MathWizardThatCanBeatYou
Yesterday at 11:01 PM
lpieleanu
Yesterday at 11:44 PM
Fractions and reciprocals
adihaya   35
N Apr 25, 2025 by Ilikeminecraft
Source: 2013 BAMO-8 #4
For a positive integer $n>2$, consider the $n-1$ fractions $$\dfrac21, \dfrac32, \cdots, \dfrac{n}{n-1}$$The product of these fractions equals $n$, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of $n$ for which this is possible and prove that you have found them all.
35 replies
adihaya
Feb 27, 2016
Ilikeminecraft
Apr 25, 2025
Fractions and reciprocals
G H J
G H BBookmark kLocked kLocked NReply
Source: 2013 BAMO-8 #4
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adihaya
1632 posts
#1 • 5 Y
Y by GoJensenOrGoHome, itslumi, Adventure10, cubres, NicoN9
For a positive integer $n>2$, consider the $n-1$ fractions $$\dfrac21, \dfrac32, \cdots, \dfrac{n}{n-1}$$The product of these fractions equals $n$, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of $n$ for which this is possible and prove that you have found them all.
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math90
1476 posts
#2 • 7 Y
Y by MSTang, mathcrazymj, Adventure10, Mango247, ItsBesi, cubres, kiyoras_2001
The answer is all perfect squares greater than $1$.
To make the product $1$, the product of the numerators should be equal to the product of the denominators. I.E, $\prod_{k=2}^{n}k(k-1)=(n-1)!^2n$ should be a perfect square. Thus $n$ should be a perfect square.
For perfect squares construct the following example:
$\prod_{i=2}^{\sqrt{n}}\frac{i-1}{i}\cdot\prod_{i=\sqrt{n}+1}^{n}\frac{i}{i-1}=\frac{1}{\sqrt{n}}\cdot\sqrt{n}=1$.
This post has been edited 1 time. Last edited by math90, May 5, 2023, 6:46 PM
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IndoMathXdZ
691 posts
#3 • 2 Y
Y by Adventure10, cubres
Maybe just my opinion, but this is a nice ez problem, with a nice idea.
I claim that all square numbers greater than $1$ satisfy.
To see that this is necessary, notice that to have that the multiply of all the fractions equal to 1, then we must have the multiply of all the fraction we have having the same denominator and numerator, which means that the product of all numerators and denominators must be a perfect square. But we know that the product is $(n-1)!^2 n$. So, $n$ must be a perfect square.

To see that this is sufficient. Notice that we have
\[ \prod_{i = 2}^{n} \frac{i}{i - 1} = n \]initially, which is a perfect square. We wanted to reverse several fractions such that the final product is $1$.
This means that we need to find several fractions which product to $\sqrt{n}$.
But this is obvious, as
\[ \prod_{i = 2}^{\sqrt{n}} \frac{i}{i - 1} = \sqrt{n} \]
This post has been edited 1 time. Last edited by IndoMathXdZ, Aug 1, 2019, 3:20 PM
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pad
1671 posts
#4 • 3 Y
Y by Adventure10, Mango247, cubres
We claim that $n$ must be a square. A construction for $n=k^2$ is
\[ \left(\frac12\cdot \frac23\cdot \frac34\cdots \cdot \frac{k-1}{k} \right) \left( \frac{k+1}{k}\cdot \frac{k+2}{k+1}\cdots \cdot \frac{n}{n-1} \right) = \frac{n}{k^2}=1. \]Suppose that $n$ works. In the final list of fractions which have product 1, the product of all the numerators must equal the product of all the denominators. This means that the product of all the numbers overall must be a square. The product of all the numbers is
\[ (2\cdot 1)(3\cdot 2)\cdots (n(n-1)) = (n-1)!^2n,\]which means $n$ must be a square.
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Eyed
1065 posts
#5 • 2 Y
Y by pad, cubres
Define a fraction as "good" if it's square root is rational, and "bad" otherwise. Observe that two good fractions multiplied together results in a good fraction and a bad and a good fraction multiplied together results in a bad fraction.

If $n = k^{2}$, we can flip the first $k-1$ fractions. If $n \neq k^{2}$, then $\frac{2}{1} \cdot\frac{3}{2}\cdot\ldots \frac{n}{n-1}$ is bad. Whenever we flip a fraction $\frac{p}{p-1} \Rightarrow \frac{p-1}{p}$, we multiply the original product by $\frac{(p-1)^{2}}{p^{2}}$, a good fraction. Then, since $1$ is a good fraction, and we start with a bad fraction, it can not be made into $1$. The only possible values of $n$ is $n = k^{2}$.
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mathlogician
1051 posts
#6 • 1 Y
Y by cubres
I claim that the problem is true iff $n$ is a perfect square.

Suppose that $n = a^2$ for some positive integer $a \geq 2$. Now we can turn this expression into $1$ by simply flipping the first $a-1$ fractions.

Now, let a fraction be called a ssquare if, when written in simplest terms, the numerator and denominator are perfect squares. Note that if the $n$ is by definition, not a ssquare. Furthermore, note that with each operation it is impossible to turn a non-ssquare into a ssquare, so we are done.
This post has been edited 1 time. Last edited by mathlogician, Jul 22, 2020, 11:43 PM
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Vitriol
113 posts
#7 • 1 Y
Y by cubres
Rephrase reciprocation as multiplying a value of the form $\left( \tfrac{k}{k+1} \right)^2$ for distinct values of $1 \le k < n$.

Now it is clear that $n$ must be a square, for the product will always be $n$ times the square of a rational by telescoping. To show that this always works, simply multiply the values
\[ \left( \frac{1}{2} \right)^2 \cdot \left( \frac{2}{3} \right)^2 \cdot \dots \cdot \left( \frac{\sqrt{n}-1}{\sqrt{n}} \right)^2 = \frac1{n}\]to $n$ to obtain $1$. $\blacksquare$
This post has been edited 1 time. Last edited by Vitriol, Aug 12, 2020, 5:54 AM
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Grizzy
920 posts
#8 • 1 Y
Y by cubres
Solution
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brianzjk
1201 posts
#9 • 1 Y
Y by cubres
We claim the only numbers that work are perfect squares.

If $n$ is a perfect square, then we can use the construction
\[\frac{1}{2}\cdot\frac{2}{3}\dots\frac{\sqrt{n}-1}{\sqrt{n}}\cdot\frac{\sqrt{n}+1}{\sqrt{n}}\cdot\frac{\sqrt{n}+2}{\sqrt{n}+1}\dots\frac{n}{n-1}\]And it is easy to see that this works.

We now want to show that $n$ has to be a perfect square for all possible values of $n$. Assume that it is possible to reciprocate some of the fractions such that the product is equal to $n$. We divide
\[\frac{2}{1}\cdot\frac{3}{2}\dots\frac{n}{n-1}=n\]by the construction that works to get
\[\left(\frac{2}{1}\right)^{1+\epsilon_1}\cdot\left(\frac{3}{2}\right)^{1+\epsilon_2}\dots\left(\frac{n}{n-2}\right)^{1+\epsilon_n}\]Where $\epsilon_i$ is either $1$ or $-1$. Then, the value of $\epsilon_i+1$ must be an even number, so $n$ must be a perfect square, as desired.
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HrishiP
1346 posts
#10 • 1 Y
Y by cubres
The answer is all $n$ such that $n$ is a perfect square greater than $1$.To see this works, we can take
$$\left[\frac{1}{2} \cdot \frac{2}{3} \dotsm\frac{\sqrt{n}-1}{\sqrt{n}}\right] \times \left[\frac{\sqrt{n}+1}{\sqrt{n}} \dotsm \frac{n}{n-1}\right].$$All of the first bracket simplifies to $\tfrac{1}{n}$ an the second to $n,$ and $\tfrac{1}{n} \times n = 1.$

To show this is the only solution, note the numerator $p$ has product $n!$ and the denominator $q$ has product $(n-1)!,$ so $pq = (n-1)!^2 \cdot n.$ Also note that for the value to be $1$, we must have $p=q$. Thus, because $pq= (n-1)!^2 \cdot n, $ $n$ must be a perfect square. $\blacksquare$
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rafaello
1079 posts
#11 • 2 Y
Y by Mango247, cubres
Posting for storage.
The problem falls after noticing the construction.

If $n$ is perfect square, then following construction works:
$$\dfrac21, \dfrac32, \cdots, \dfrac{\sqrt{n}}{\sqrt{n}-1}, \dfrac{\sqrt{n}}{\sqrt{n}+1}, \cdots, \dfrac{n-1}{n}.$$
Now we claim that $n$ must be a perfect square.
Notice that for all $p\in\mathbb{P}$, we must have $$2\mid \nu_p(n!)+\nu_p((n-1)!),$$since otherwise for some prime, we have a number of those primes odd and thus we cannot group them into two groups having the same cardinality.
Rewriting we obtain, $$2\mid \nu_p(n)+2\nu_p((n-1)!)\implies \nu_p(n)=2k,$$hence $n$ must be a perfect square in order to form such product.
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IAmTheHazard
5001 posts
#12 • 2 Y
Y by centslordm, cubres
Let $a$ be the product of the fractions not flipped and $b$ the product of the fractions which are flipped. Then we have $ab=n$ and $\tfrac{a}{b}=1$, implying $a^2=n$. Since $a$ is rational it follows that $n$ must be a perfect square. On the other hand, if $n=m^2$, we can flip $\tfrac{2}{1},\tfrac{3}{2},\ldots,\tfrac{m}{m-1}$.
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Afo
1002 posts
#13 • 1 Y
Y by cubres
Solution. The answer is all perfect squares $> 2$. Note that the product of all the numerators and the denominators must have even powers so $n!(n-1)! = (n-1)!^2 \times n$ is a perfect square and so is $n$. The construction is to divide into $\sqrt{n}+1$ groups of $\sqrt{n}-1$ fractions and reciprocate every group except the first. Here's an example for $n = 9$.
$$\left( \frac{2}{1} \cdot \frac{3}{2} \right)\cdot \left( \frac{4}{3} \cdot \frac{5}{4} \right)^{-1}\cdot \left( \frac{6}{5} \cdot \frac{7}{6} \right)^{-1}\cdot \left( \frac{8}{7} \cdot \frac{9}{8} \right)^{-1} = 1$$
This post has been edited 1 time. Last edited by Afo, Sep 6, 2021, 9:54 AM
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HamstPan38825
8860 posts
#14 • 1 Y
Y by cubres
The answer is $n=k^2$ for some positive integer $k>1$. Upon reciprocating any fraction, the product is multiplied by the perfect square of a rational number. As a result, the original product must also have been a perfect square by invariance.

Furthermore, for $n = k^2$, simply reciprocate the first $k-1$ fractions (which multiply to $k$). Then the product is multiplied by $\frac 1{n^2}$, and thus it will equal 1.
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Mogmog8
1080 posts
#15 • 2 Y
Y by centslordm, cubres
If $n=a^2$, we flip $\frac{2}{1}$, $\frac{3}{2}$, $\dots$, $\frac{a}{a-1}$ over and our product becomes \[\underbrace{\frac{1}{a-1}}_{\text{first } a-2\text{ terms}}\cdot\underbrace{\frac{1}{a+1}}_{\text{terms }a+1\text{ to }a^2-1}\cdot\frac{a-1}{a}\cdot\frac{a+1}{a}\cdot\frac{a^2}{a^2-1}=1\]$\square$
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peppapig_
281 posts
#16 • 6 Y
Y by Taco12, Jndd, Significant, player01, mulberrykid, cubres
We claim that only positive perfect squares work. Notice that if $n$ is not a perfect square, then we have that the total product of all integers on the top and on the bottom is not a perfect square, meaning that we can never have the top and bottom divide each other out to equal $1$. Now let $n=k^2$. We now have
\[\frac{1}{2}*\frac{2}{3}*\cdots{}*\frac{k-1}{k}=\frac{1}{k}\]and
\[\frac{k+1}{k}*\frac{k+2}{k+1}*\cdots{}*\frac{k^2}{k^2-1}=k\]
Multiplying these together gives us our final product of $1$, as desired, and we are done.
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trk08
614 posts
#17 • 1 Y
Y by cubres
When multiplying these all out in regular order, we get $n$ as given in the problem. If we take the reciprocal of some of them, let us say that there product is $x$. This means that the total sum would turn out to be $n\cdot x^2$ which we claim to be $1$. Simplifying this, we get:
\[x=\frac{1}{\sqrt{n}}.\]We also know that $x$ has to be rational meaning that $n$ has to be a perfect square which we can say is $a^2$. If we substitute this back in, we are basically saying that the original sequence is:
\[\frac21, \frac32, \cdots, \frac{a^2}{a^2-1},\]and we want a product of the reciprocal of some of these to be $\frac{1}{a}$. We know that $a<a^2$, so we can say that a possible value of $x$ could be:
\[\frac{1}{2}\cdot \frac{2}{3}\cdot \dots \frac{a-1}{a}\]\[\frac{1}{a},\]which means that $x^2\cdot n=1$.

Thus, if $n$ is a perfect square, this condition works.
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S.Das93
709 posts
#18 • 1 Y
Y by cubres
It is quite intuitive that to get a product of $1$, we need to have a product that cancels in the numerator and denominator. For these to equal the same product, $n$ would have to be the product of these - or $n=k^2$ for the product $k$, thus $n$ must be a perfect square.
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gracemoon124
872 posts
#19 • 1 Y
Y by cubres
$\textbf{Claim:}$ $n$ must be a perfect square.

$\emph{Proof:}$ Let $\{t_1, t_2, \dots, t_k\}$ be a subset of $\{2, 3, \dots, n\}$ such that all the numbers of the form $\tfrac{t_i}{t_i-1}$ are reciprocated.

Notice that reciprocating a factor $a$ of $n$ gives the resulting number as $\tfrac{n}{a^2}$. Therefore, reciprocating all the numbers of the form $\tfrac{t_i}{t_i-1}$ gives our end result as
\[\frac{n}{\left(\frac{t_1}{t_1-1}\cdot\frac{t_2}{t_2-1}\cdot\dots\cdot\frac{t_k}{t_k-1}\right)^2}=1\]implying that we require
\[\frac{t_1}{t_1-1}\cdot\frac{t_2}{t_2-1}\cdot\dots\cdot\frac{t_k}{t_k-1} =\sqrt{n}.\]Since the LHS product is rational, we require the RHS to be rational as well. But $n$ is an integer, it isn't a fraction. Hence, $n$ must be a perfect square. $\square$
$\textbf{Construction:}$ Let $n=k^2$, notice that
\[\frac{k^2}{k^2-1}\cdot\frac{k+1}{k}\cdot\frac{k-1}{k-2}\cdot\frac{k-2}{k-3}\dots\frac{3}{2}\cdot\frac{2}{1}=k.\]So the answer is $\boxed{\text{all perfect squares greater than 1}}$.
$\blacksquare$

motivation
This post has been edited 1 time. Last edited by gracemoon124, May 5, 2023, 6:31 PM
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Alcumusgrinder07
95 posts
#21 • 1 Y
Y by cubres
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ryanbear
1056 posts
#22 • 1 Y
Y by cubres
The product is equal to $n$. To flip a fraction $\frac{m}{m+1}$, multiply by $\frac{(m+1)^2}{m^2}$. Define "perfect square" as the square of a rational number. If $n$ is not a perfect square, then multiplying by a perfect square keeps the product not a perfect square. So the product can never reach $1$, which is a perfect square. So $n$ has to be a perfect square. If $n$ is a perfect square, flipping the first $\sqrt{n}-1$ fractions results in the product being $1$.
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ex-center
27 posts
#23 • 1 Y
Y by cubres
The answer is $n$ such that $n$ is a perfect square. This can be achieved by
$$\left( \frac{2}{1}\cdot \frac{3}{2} \cdot \dots \cdot\frac{\sqrt{n}}{\sqrt{n}-1} \right) \cdot \left( \frac{\sqrt{n}}{\sqrt{n}+1} \cdot \frac{\sqrt{n}+1}{\sqrt{n}+2} \cdot \dots \cdot \frac{n-2}{n-1}\right) \cdot \frac{n-1}{n} = 1$$Notice the product of the reciprocated fractions is $n$ if we flip a fraction $\frac{a}{b}$ we multiply by $\frac{b^{2}}{a^{2}}$ so each prime factor gets changed by a multiple of $2$ hence $n$ must be a square since $1$ is a square.
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joshualiu315
2534 posts
#24 • 1 Y
Y by cubres
We claim the answer is all perfect squares. A construction for $n=k^2$ is

\[ \left(\frac{1}{2}\cdot \frac{2}{3}\cdot \frac{3}{4}\cdots \cdot \frac{k-1}{k} \right) \left( \frac{k+1}{k}\cdot \frac{k+2}{k+1}\cdots \cdot \frac{n}{n-1} \right). \]
Now, we must prove that only perfect squares work. Notice that the numerator and denominator must be equal, or the product of the numerator and denominator must be a perfect square. Now, this value always stays constant throughout, so we have that

\[n!(n-1)!=n((n-1)!)^2\]
is a perfect square, implying that $n$ must be a perfect square.
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cj13609517288
1909 posts
#25 • 1 Y
Y by cubres
The answer is when $n$ is a perfect square greater than one.

If the product can be made into $1$, the product of the numerator and the denominator must be a square. Thus $(n-1)!\cdot n!$ is a square, so $n$ is a square. When $n$ is a square, we can flip over $\frac21,\frac32,\dots,\frac{\sqrt{n}}{\sqrt{n}-1}$ and win. $\blacksquare$
This post has been edited 1 time. Last edited by cj13609517288, Dec 8, 2023, 4:49 PM
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eibc
600 posts
#26 • 1 Y
Y by cubres
The answer is all $n$ which are perfect squares. Note that the product of all the numerators must be equal to the product of all of the denominators of the fractions (after some are flipped) for the product to be $1$. Hence $1 \cdot 2^2 \cdot 3^2 \cdots (n - 1)^2 \cdot n$ must be a perfect square, so $n$ is a perfect square.

For $n = k^2$, notice that
$$\frac{k^2}{k^2 - 1} \cdot \frac{k^2 - 1}{k^2 - 2} \cdots \frac{k + 1}{k} \cdot \frac{k - 1}{k} \cdot \frac{k - 2}{k - 1} \cdots \frac{1}{2} = \frac{k^2}{k \cdot k} = 1. $$
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cursed_tangent1434
623 posts
#27 • 2 Y
Y by GeoKing, cubres
We claim that the answer is any square number. Let an integer which satisfies the required condition be called good.

Claim : Any $n$ such that $v_p(n)=k$ where $k$ is odd is not good
Proof : Notice that if the numerator of a certain fraction is divisible by a certain prime, the denominator is not, and vice versa. Thus, if $v_p(n!)+v_p(n-1)!$ is an odd number it is impossible to divide it by two (such that $v_p$ of the numerator is equal to the denominator - so that the final product is 1). Now, notice that when $n=p^km$ for odd $k$, and $p \nmid m$
$$v_p(2^km!) + v_p((2^km-1)!) = 2^k + (2^k - k) = 2^{k+1} - k$$Thus, if $k$ is odd $v_p(n!)+v_p(n-1)!$, is also clearly odd. Thus, all odd numbers are not good as claimed.

Now, we will show that sqaure numbers are in fact good.

Algorithm : All squares numbers $n$ are clearly good, by the following algorithm.
Simply reciprocate the first $k-1$ fractions. Thus, we obtain,
\[\frac{1}{2}\cdot \frac{2}{3} \cdots \frac{k-1}{k}\cdot \frac{k+1}{k} \cdot \frac{k+2}{k+1} \cdots \frac{k^2}{k^2-1}\]
Proof : Clearly, this product $P = \frac{k^2}{k\cdot k} = 1$, thus all squares obviously work.
This post has been edited 3 times. Last edited by cursed_tangent1434, Dec 29, 2023, 4:09 AM
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shendrew7
795 posts
#28 • 1 Y
Y by cubres
The boolean "product is the square of a rational" is invariant. Thus $n$ must be a $\boxed{\text{perfect square}}$. Our construction is to flip the first $\sqrt{n}-1$ fractions. $\blacksquare$
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AshAuktober
1005 posts
#29 • 1 Y
Y by cubres
Note that flipping any of the fractions is equivalent to dividing the product by a rational square, so if $n$ is a value such that the product can be $1$, then $n$ must be a perfect square $>1$. But note that all such n work; indeed, let $n = k^2$.Then we can flip the fractions $\frac{2}{1}, \cdots, \frac{k}{k-1}$. This divides the original product by $k^2 = n$, so the final product becomes $1$, which is what we wanted. $\square$
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lpieleanu
3001 posts
#30 • 1 Y
Y by cubres
Solution
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eg4334
637 posts
#31 • 1 Y
Y by cubres
First off, considering any prime $p$ after we flip it the $v_p$ of the numerator is equal to the $v_p$ of the denominator, hence the product of the numerator and denominator is a square, i.e: $$n! \cdot (n-1)! \text{ = square} \implies n \text{ = square}$$I claim $\boxed{\text{all squares}}$ work. We use induction. $2^2$ case is obvious. If the result is true for $k^2$, then to construct $(k+1)^2$ look at the final $2k+1$ terms because the first $k^2$ already work. Out of the last $2k+1$ terms, flip the last $k+1$. Keep the rest the same. The product of the last $2k+1$ then turns into $$\frac{k^2+k}{k^2} \cdot \frac{k^2+k}{(k+1)^2} = 1$$, done.
This post has been edited 1 time. Last edited by eg4334, Dec 6, 2024, 3:07 AM
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Benbenwang
548 posts
#32 • 1 Y
Y by cubres
We claim that all perfect squares $n \ge 4$ work.

Let the product of the fractions we wish to reciprocate be $\frac{a}{b}$ and the product of the remaining fractions be $k$. Clearly we have that:
$$k\cdot\frac{a}{b} = n  \:\:\: \text{and} \:\:\: k\cdot\frac{b}{a} = 1.$$Dividing equations yields
$$n = \left(\frac{a}{b}\right)^2$$since $n$ is an integer it must be a perfect square.

It is also easy to see that the first $\sqrt{n}-1$ terms multiply to $\sqrt{n}$, and as a result the last $n-\sqrt{n}$ terms also multiply to $\sqrt{n}.$ Reciprocate either the first $\sqrt{n}-1$ terms or last $n-\sqrt{n}$ terms and we are done. $\square$
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Maximilian113
575 posts
#33 • 1 Y
Y by cubres
Suppose that we reciprocate the fractions $a_1, a_2, \cdots, a_k,$ which should obviously be distinct. Then we have that $\frac{n}{(a_1a_2 \cdots a_k)^2} = 1 \implies n = (a_1a_2 \cdots a_k)^2.$ Thus $n$ is a perfect square.

Now, we show that all perfect squares greater than $1$ work. For the construction, consider $a_1=\frac{2}{1}, a_2 = \frac{3}{2}, \cdots a_k = \frac{\sqrt{n}}{\sqrt{n}-1}.$
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sansgankrsngupta
143 posts
#34 • 1 Y
Y by cubres
OG! The answer is all perfect squares $n$
Construction:
Let $n=k^2$, reciprocate the fractions $$ \frac{2}{1}, \frac{3}{2} \ldots \frac{k}{k-1}$$Proof of necessity:
$$P= \frac{2}{1}* \frac{3}{2} \cdots * \frac{n}{n-1}=n$$Assume you reciprocate some of the fractions among the fractions to get 1 .
We view reciprocation of a fraction $f$ as $\frac{1}{f}=f.\frac{1}{f^2}$. Hence we are essentially multiplying $P$ by the square of some rational number.
Hence we get $Pq^2=nq^2=1 \iff n= (\frac{1}{q})^2$, where $q$ is a rational number. so $n$ is also the square of some rational number.
$n= \frac{p^2}{q^2}$ where $(p,q)=1$ and $p$ and $q$ are integers since $n$ is an integer we have that $p^2|q^2$ but we know that $gcd(p^2,q^2)=1$. Thus, $p^2=1$. Hence $n=p^2$ as desired.
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gladIasked
648 posts
#35
Y by
The answer is all $n=k^2$.

Let $c=p_1^{e_1}p_2^{e_2}\cdots$ be the ``prime factorization" of the product of (possibly reciprocated) fractions, where each $e_i$ is a not necessarily positive integer. Clearly, the parity of each $e_i$ is invariant no matter how we reciprocate the fractions. Because $1=p_1^0p_2^0\cdots$, we know that $n$ must be a perfect square (as the product of the fractions can equal $n$).

When $n=k^2$, it's easy to see that reciprocating the first $k-1$ fractions yields a product of $1$, as desired. $\blacksquare$
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de-Kirschbaum
198 posts
#36
Y by
We claim that we can do this for all $n=k^2>2$. Note that each time we flip a fraction $\frac{a}{b}$ it is equivalent to multiplying it by $\frac{b^2}{a^2}$. So this is saying that for some $S^2$ where $S \in \mathbb Q$, representing the product of all the inverses of fractions we are flipping, we have $nS^2=1 \implies n=(\frac{1}{S})^2$. Since $n \in \mathbb Z$ we must have $\frac{1}{S} \in \mathbb Z$ and thus $n$ is a perfect square.

Now we will flip every fraction up until $\frac{\sqrt{n}}{\sqrt{n}-1}$ and keep the rest the same. Then we have $$\frac{1}{2}\frac{2}{3}\frac{3}{4}\cdots\frac{\sqrt{n}-2}{\sqrt{n}-1}\frac{\sqrt{n}-1}{\sqrt{n}}\frac{\sqrt{n}+1}{\sqrt{n}}\cdots\frac{n}{n-1}=\frac{n}{\sqrt{n}\sqrt{n}}=1$$
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Ilikeminecraft
617 posts
#37
Y by
I claim that all perfect squares $n > 1$ are valid. Let $n = k^2.$ Thus, $1$ can be made from: $$\frac12 \cdot \frac23 \cdot \ldots \cdot \frac{k - 1}{k} \cdot \frac{k + 1}{k} \cdot \ldots \cdot \frac{k^2}{k^2-1} = 1$$Now, I claim that if $n$ isn't a perfect square, then it won't work. Notice that flipping a fraction $\frac c{c-1}$ will just multiply the current value by the square number $\frac{(c-1)^2}{c^2}.$ Thus, since $n$ isn't a perfect square, you can never achieve another perfect square, $1.$
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