Plan ahead for the next school year. Schedule your class today!

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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

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0 replies
jwelsh
Jul 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
Two Recurrences
rkm0959   16
N 10 minutes ago by MathLuis
Source: 2015 Final Korean Mathematical Olympiad Day 2 Problem 5
For a fixed positive integer $k$, there are two sequences $A_n$ and $B_n$.
They are defined inductively, by the following recurrences.
$A_1 = k$, $A_2 = k$, $A_{n+2} = A_{n}A_{n+1}$
$B_1 = 1$, $B_2 = k$, $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$
Prove that for all positive integers $n$, $A_{2n}B_{n+3}$ is an integer.
16 replies
rkm0959
Mar 22, 2015
MathLuis
10 minutes ago
Greatest algebra ever
EpicBird08   23
N 13 minutes ago by ryanbear
Source: ISL 2024/A2
Let $n$ be a positive integer. Find the minimum possible value of
\[
S = 2^0 x_0^2 + 2^1 x_1^2 + \dots + 2^n x_n^2,
\]where $x_0, x_1, \dots, x_n$ are nonnegative integers such that $x_0 + x_1 + \dots + x_n = n$.
23 replies
1 viewing
EpicBird08
Jul 16, 2025
ryanbear
13 minutes ago
A Sequence of +1's and -1's
ike.chen   37
N 22 minutes ago by quantam13
Source: ISL 2022/C1
A $\pm 1$-sequence is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
37 replies
ike.chen
Jul 9, 2023
quantam13
22 minutes ago
Bonza functions
KevinYang2.71   69
N 25 minutes ago by monval
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
69 replies
KevinYang2.71
Jul 15, 2025
monval
25 minutes ago
Bashtastic
nosysnow   60
N 2 hours ago by iwastedmyusername
Source: 2018 AIME 1 #11
Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.
60 replies
nosysnow
Mar 7, 2018
iwastedmyusername
2 hours ago
specific topic studying
Samcool   3
N 2 hours ago by harshu13
I was wondering if there are any reasources that i could get practice porblems for specific topics like Number Theory or maybe even more specific like Mod or Diophante Equations etc sop I can pratice on these specific areas since I know what topics im not the best in.
3 replies
Samcool
Yesterday at 4:07 AM
harshu13
2 hours ago
djmathman mock 2013 p12
alcumusftwgrind   5
N 4 hours ago by alcumusftwgrind
Source: djmathman mock 2013 p12
whats wrong with my solution?
let $a=\log y$ and $b= \log x$ so $a/b=6$ and $(\log2+a)/(\log2+b)=5\implies \log 2+a=5\log 2+5b\implies a=4\log 2+5b \implies 6b=4\log2+5b\implies b=4\log 2 $

therefore $a=24\log2$

our desired expression is $(\log 4+a)/(\log4+b)=(2 \log 2 +a) / (2 \log 2 +b)= 26/6=13/3$
5 replies
1 viewing
alcumusftwgrind
Monday at 9:56 PM
alcumusftwgrind
4 hours ago
AMC 10/12 trainer
grapecoder   12
N 4 hours ago by clod
Hey guys, I created an AMC 8/10/12 trainer a while back which has a bunch of different resources. It saves statistics and has multiple modes, allowing you to do problems in an alcumus style or full exam mode with a timer and multiple solutions scraped from the AOPS wiki. If anyone's interested, I can work on adding AIME and more.

Here's the link: https://amctrainer.org
And here's the code (if anyone's interested): https://github.com/megagames-me/amc-trainer

Any feedback/suggestions are appreciated!
12 replies
grapecoder
Oct 22, 2023
clod
4 hours ago
Math Tests :(
Benq   74
N 4 hours ago by smaran
Source: 2017 AMC 10B #25 / 2017 AMC 12B #21
Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?

$\textbf{(A)} \text{ 92} \qquad \textbf{(B)} \text{ 94} \qquad \textbf{(C)} \text{ 96} \qquad \textbf{(D)} \text{ 98} \qquad \textbf{(E)} \text{ 100}$
74 replies
Benq
Feb 16, 2017
smaran
4 hours ago
9 Most recommended resource
HiCalculus   24
N 4 hours ago by HiCalculus
For this poll, I'm going to let everyone choose 1 option to vote for so I can see which resource is the absolute best. Later, I'll make another poll with 2 options per user to find the best way to incorporate 2 resources to study for the test in around 3 months from now.
24 replies
HiCalculus
Jul 27, 2025
HiCalculus
4 hours ago
AoPS Classes for AMC 10
Existing_Human1   7
N 5 hours ago by HiCalculus
Hello community!

I have never taken an aops class before, so I'm wondering which one I should take as my first. Should I take one of the contest prep ones (like the problem series or final fives), or just the standard aops textbook-driven ones. I'm hoping to at least qualify for AIME next year, and my Mathcounts scores are 43 for chapter (good for me) and a 34 on state (bad for me).
7 replies
Existing_Human1
Jun 17, 2025
HiCalculus
5 hours ago
Hexagonal Prison
franchester   38
N Yesterday at 8:59 PM by martianrunner
Source: 2018 AIME I #7
A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).
38 replies
franchester
Mar 7, 2018
martianrunner
Yesterday at 8:59 PM
Prove Collinearity
tc1729   132
N Yesterday at 8:59 PM by cubres
Source: 2012 USAMO Day 2 #5 and USAJMO Day 2 #6
Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.
132 replies
tc1729
Apr 25, 2012
cubres
Yesterday at 8:59 PM
hardest subject on amc 10?
hgmium   30
N Yesterday at 4:01 PM by OGMATH
amc 10 is coming in a few months, and I'm not sure what to spent most of my time on
what topic should i focus on the most?
30 replies
hgmium
Jun 26, 2025
OGMATH
Yesterday at 4:01 PM
Computing functions
BBNoDollar   8
N May 24, 2025 by wh0nix
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
8 replies
BBNoDollar
May 18, 2025
wh0nix
May 24, 2025
Computing functions
G H J
G H BBookmark kLocked kLocked NReply
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BBNoDollar
15 posts
#1
Y by
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
Z K Y
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alinazarboland
172 posts
#2
Y by
Here's a sketch of a method which solves every single mobius tranform problem I saw.
Let $z_1,z_2$ be the two complex roots of $f(z)=z$. Then, since a mobius transform is just a combination of shifting,scaling,rotating, and inversion, for any complex number $z$ we have:
$$(z , \infty ; z_1,z_2) = (f(z) , \frac{a}{c} ; z_1,z_2)$$If you write this $n$ times you'd get:
$$k^n .\frac{z-z_1}{z-z_2} = \frac{f_n - z_1}{f_n - z_2}$$Where $k = \frac{a/c - z_1}{a/c - z_2}$.Now let $f_n(x) = \frac{x}{1 + nx}$ for some $n$. One can easily get $k^n=1$(by comparing the coefficient of $x$ in the respective polynomial identity) and so $x_1=x_2$(comparing $x^2$s).
Now, $x_1=x_2$ means we have a double root for $f(x)=x$ and delta=0 so $(d-a)^2+4bc=0$. Combining with the fact that $x_1,x_2$ are fix points of every $f_k$ , we'll get $(n-1)^2+0=0$ and $n=1$
This post has been edited 2 times. Last edited by alinazarboland, May 18, 2025, 7:42 PM
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alinazarboland
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Here are two old problems one from $2012$ IMC and one from Iranian Olympiad which are trivial with this method
https://artofproblemsolving.com/community/c7h491145p2754513
https://artofproblemsolving.com/community/c6h368215p2026678
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BBNoDollar
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alinazarboland wrote:
Here's a sketch of a method which solves every single mobius tranform problem I saw.
Let $z_1,z_2$ be the two complex roots of $f(z)=z$. Then, since a mobius transform is just a combination of shifting,scaling,rotating, and inversion, for any complex number $z$ we have:
$$(z , \infty ; z_1,z_2) = (f(z) , \frac{a}{c} ; z_1,z_2)$$If you write this $n$ times you'd get:
$$k^n .\frac{z-z_1}{z-z_2} = \frac{f_n - z_1}{f_n - z_2}$$Where $k = \frac{a/c - z_1}{a/c - z_2}$.Now let $f_n(x) = \frac{x}{1 + nx}$ for some $n$. One can easily get $k^n=1$(by comparing the coefficient of $x$ in the respective polynomial identity) and so $x_1=x_2$(comparing $x^2$s).
Now, $x_1=x_2$ means we have a double root for $f(x)=x$ and delta=0 so $(d-a)^2+4bc=0$. Combining with the fact that $x_1,x_2$ are fix points of every $f_k$ , we'll get $(n-1)^2+0=0$ and $n=1$

Thank you very much, i appreciate this solution ! I can understand it, but i need a 9th grade solution. I solved the ''reciprocal'' implication by induction, now i need to demonstrate the ''direct'' one. Can you or anyone help me ?
This post has been edited 1 time. Last edited by BBNoDollar, May 18, 2025, 10:15 PM
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ICE_CNME_4
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Bumping this
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ICE_CNME_4
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Bump. Bump
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BBNoDollar
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BUMPING for 9th grade solution
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ICE_CNME_4
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Someone for this?
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wh0nix
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#9
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Hint
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