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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 lengths are equal
62861   32
N a minute ago by OronSH
Source: IMO 2015 Shortlist, G5
Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.

Proposed by El Salvador
32 replies
62861
Jul 7, 2016
OronSH
a minute ago
Number Theory Marahon
Jupiterballs   15
N 4 minutes ago by ItzsleepyXD
Let's start a number theory marathon
Rules:-
just don't post >2 problems before a solution and be friendly :)

I'll start
P1
15 replies
Jupiterballs
Jun 23, 2025
ItzsleepyXD
4 minutes ago
bounded or all n?
X.Allaberdiyev   2
N 12 minutes ago by young_desi
Source: IMSC Problem 4
Determine all $n$ such that it is possible to find a convex $n$-gon which can be tiled with triangles having angles $15^\circ$, $75^\circ$ and $90^\circ$.
2 replies
1 viewing
X.Allaberdiyev
Jul 7, 2025
young_desi
12 minutes ago
Parallel lines (extension of previous problem)
RANDOM__USER   0
25 minutes ago
Source: Own
Let \(D\) be an arbitrary point on the side \(BC\) in a triangle \(\triangle{ABC}\). Let \(E\) and \(F\) be the intersection of the lines parallel to \(AC\) and \(AB\) through \(D\) with \(AB\) and \(AC\). Let \(G\) be the intersection of \((AFE)\) with \((ABC)\). Let \(M\) be the midpoint of \(BC\) and \(X\) the intersection of \(AM\) with \((ABC)\). Let \(H\) be the intersection of \((XMG)\) with \(BC\). Prove that \(EF\) is parallel to \(AH\).

IMAGE

Note: This is another property of a configuration I posted before where one needed to prove that \(X, D\) and \(G\) are collinear. There are surprisingly many properties in the configuration posted earlier :P
0 replies
RANDOM__USER
25 minutes ago
0 replies
Qualifying for AIME through AMC 10
AdrienMarieLegendre   0
3 hours ago
hello..!!
i am a sophomore, and my goal is to qualify for AIME this year and MP4G; I am also reaching towards getting hr in amc 10.
I have competition experience, but I never really did that well (SMT, HMMT, local comps). I got a 55 on the AMC 10 last year with minimal prep.
my current strategy to prepare is doing AoPS vol 1, intro to counting and probability, and intro to number theory, as well as doing mocks, alcumus, and math dash. I struggle with timing and have lots of content gaps, so on amc mocks i can only solve ~5-10 questions and score ~50-60. i also struggle with memorizing the theory that I learn.
What should I do to improve? Is ~4 months enough to prepare? (I've been studying for around a month now, and I am on chapter 16 of AoPS Vol 1. )
0 replies
AdrienMarieLegendre
3 hours ago
0 replies
USAJMO problem 3: Inequality
BOGTRO   111
N 4 hours ago by InftyByond
Let $a,b,c$ be positive real numbers. Prove that $\frac{a^3+3b^3}{5a+b}+\frac{b^3+3c^3}{5b+c}+\frac{c^3+3a^3}{5c+a} \geq \frac{2}{3}(a^2+b^2+c^2)$.
111 replies
BOGTRO
Apr 24, 2012
InftyByond
4 hours ago
Isogonal Conjugates: 2011 USAMO #5
tenniskidperson3   78
N 5 hours ago by littlefox_amc
Let $P$ be a given point inside quadrilateral $ABCD$. Points $Q_1$ and $Q_2$ are located within $ABCD$ such that
\[\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.\] Prove that $\overline{Q_1Q_2}\parallel\overline{AB}$ if and only if $\overline{Q_1Q_2}\parallel\overline{CD}$.
78 replies
tenniskidperson3
Apr 28, 2011
littlefox_amc
5 hours ago
Resources for algebra and combinatorics and geometry
Not__Infinity   4
N 6 hours ago by Not__Infinity
So yeah, could just anyone recommend me of any resources like book/website/test paper pyq or anything else. Also, you can give me resources of any level u want. Just label it like AMC, usamo, imo or imo tst. I would like to hear that.

Edit- is self studying for hard olympiad like usamo level worth it? Only just self studying and some random getting random to understand an concept?
4 replies
Not__Infinity
Yesterday at 3:10 PM
Not__Infinity
6 hours ago
For a 170 AMC 10 index: OTIS, WOOT, or something else
AD314159   2
N Today at 12:19 AM by Andyluo
My goal is to make USAJMO this year (final year eligible), I got a 118.5 on the AMC 10 and a 5 on AIME in 2025 (AMC 10 made 2 sillies, AIME I really couldn't get higher than a 5). This seems ambitious (esp given 230 cutoff this year), but I'm willing to dedicate 15-20 hours a week to math, consistently, and am currently doing Volume 2 and past papers. I've heard that both WOOT and OTIS are great programs, but I've also received conflicting advice on which one would be better suited for my current level and goals.
2 replies
AD314159
Yesterday at 10:42 PM
Andyluo
Today at 12:19 AM
geo equals ForeBoding For Dennis
dchenmathcounts   105
N Yesterday at 11:39 PM by mudkip42
Source: USAJMO 2020/4
Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \perp AC$ and $EF \parallel BC$. Prove that $FB = FD$.

Milan Haiman
105 replies
dchenmathcounts
Jun 21, 2020
mudkip42
Yesterday at 11:39 PM
Isosceles Triangulation
worthawholebean   72
N Yesterday at 8:10 PM by lpieleanu
Source: USAMO 2008 Problem 4
Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n - 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $ \mathcal{P}$ into $ n - 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.
72 replies
worthawholebean
May 1, 2008
lpieleanu
Yesterday at 8:10 PM
2016 Sets
NormanWho   115
N Yesterday at 7:44 PM by cxsmi
Source: 2016 USAJMO 4
Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set ${1, 2,...,N}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.
115 replies
NormanWho
Apr 20, 2016
cxsmi
Yesterday at 7:44 PM
Excircle
62861   75
N Yesterday at 7:01 PM by eg4334
Source: JMO 2019 Problem 4, by Ankan Bhattacharya, Zack Chroman, and Anant Mudgal
Let $ABC$ be a triangle with $\angle ABC$ obtuse. The $A$-excircle is a circle in the exterior of $\triangle ABC$ that is tangent to side $BC$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respectively. Can line $EF$ be tangent to the $A$-excircle?

Proposed by Ankan Bhattacharya, Zack Chroman, and Anant Mudgal
75 replies
62861
Apr 18, 2019
eg4334
Yesterday at 7:01 PM
'10 USAMO #2: Students in a Circle
1=2   53
N Yesterday at 5:42 PM by fearsum_fyz
There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.
53 replies
1=2
Apr 29, 2010
fearsum_fyz
Yesterday at 5:42 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 $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
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alinazarboland
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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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