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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
Interesting inequalities
sqing   13
N 4 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , (a+k )(b+c)=k+1.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{2k-3+2\sqrt{k+1}}{3k-1}$$Where $ k\geq \frac{2}{3}.$
Let $ a,b,c\geq 0 , (a+1)(b+c)=2.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq 2\sqrt{2}-1$$Let $ a,b,c\geq 0 , (a+3)(b+c)=4.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{7}{4}$$Let $ a,b,c\geq 0 , (3a+2)(b+c)= 5.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{2(2\sqrt{15}-5)}{3}$$
13 replies
sqing
May 10, 2025
sqing
4 minutes ago
Inspired by 2022 MARBLE - Mock ARML
sqing   2
N 7 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}= 5 $ and $ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32. $ Prove that $$\frac{3}{2}>ab+bc+ca \geq  \frac{49}{34}$$Let $ a,b,c\geq 0 ,ab+bc+ca = \frac{49}{34} $ and $ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32. $ Prove that $$\frac{51}{10}>\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\geq5$$Let $ a,b,c\geq 0 ,ab+bc+ca = \frac{49}{34} $ and $ \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=5. $ Prove that $$\frac{63}{2}<\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\leq32$$
2 replies
sqing
Yesterday at 1:34 PM
sqing
7 minutes ago
Beijing High School Mathematics Competition 2025 Q1
SunnyEvan   3
N 15 minutes ago by sqing
Let $ a,b,c,d \in R^+ $. Prove that:
$$ \frac{1}{a^4+b^4+c^4+abcd}+\frac{1}{b^4+c^4+d^4+abcd}+\frac{1}{c^4+d^4+a^4+abcd}+\frac{1}{d^4+a^4+b^4+abcd} \leq \frac{1}{abcd} $$
3 replies
SunnyEvan
Yesterday at 6:10 AM
sqing
15 minutes ago
Transposition?
EeEeRUT   2
N 25 minutes ago by Bluecloud123
Source: Thailand MO 2025 P8
For each integer sequence $a_1, a_2, a_3, \dots, a_n$, a single parity swapping is to choose $2$ terms in this sequence, say $a_i$ and $a_j$, such that $a_i + a_j$ is odd, then switch their placement, while the other terms stay in place. This creates a new sequence.

Find the minimal number of single parity swapping to transform the sequence $1,2,3, \dots, 2025$ to $2025, \dots, 3, 2, 1$, using only single parity swapping.
2 replies
EeEeRUT
May 14, 2025
Bluecloud123
25 minutes ago
An FE. Who woulda thunk it?
nikenissan   118
N 3 hours ago by maromex
Source: 2021 USAJMO Problem 1
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b,$ \[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]
118 replies
nikenissan
Apr 15, 2021
maromex
3 hours ago
high tech FE as J1?!
imagien_bad   62
N May 16, 2025 by jasperE3
Source: USAJMO 2025/1
Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective.
Note: A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.
62 replies
imagien_bad
Mar 20, 2025
jasperE3
May 16, 2025
Nonlinear System
worthawholebean   39
N May 12, 2025 by GeoKing
Source: AIME 2010I Problem 9
Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 - xyz = 2$, $ y^3 - xyz = 6$, $ z^3 - xyz = 20$. The greatest possible value of $ a^3 + b^3 + c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m + n$.
39 replies
worthawholebean
Mar 17, 2010
GeoKing
May 12, 2025
Sequence of integers
tenniskidperson3   32
N May 12, 2025 by N3bula
Source: 2012 USAMO problem #3
Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \ldots$ of nonzero integers such that the equality \[a_k+2a_{2k}+\ldots+na_{nk}=0\]holds for every positive integer $k$.
32 replies
tenniskidperson3
Apr 24, 2012
N3bula
May 12, 2025
calc bash p15
Sleepy_Head   8
N May 4, 2025 by Tetra_scheme
Source: 2025 AIME II p15
Let
\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.
8 replies
Sleepy_Head
Feb 13, 2025
Tetra_scheme
May 4, 2025
System
worthawholebean   10
N Apr 29, 2025 by daijobu
Source: AIME 2008II Problem 14
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m+n$.
10 replies
worthawholebean
Apr 3, 2008
daijobu
Apr 29, 2025
Sort of additive function
tenniskidperson3   112
N Apr 27, 2025 by anudeep
Source: 2015 USAJMO problem 4
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that\[f(x)+f(t)=f(y)+f(z)\]for all rational numbers $x<y<z<t$ that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)
112 replies
tenniskidperson3
Apr 29, 2015
anudeep
Apr 27, 2025
Inequality on a Quartic
patrickhompe   136
N Apr 25, 2025 by Ilikeminecraft
Source: USAMO 2014, Problem 1
Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.
136 replies
patrickhompe
Apr 29, 2014
Ilikeminecraft
Apr 25, 2025
average FE
KevinYang2.71   81
N Apr 25, 2025 by Ilikeminecraft
Source: USAJMO 2024/5
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]for all $x,y\in\mathbb{R}$.

Proposed by Carl Schildkraut
81 replies
KevinYang2.71
Mar 21, 2024
Ilikeminecraft
Apr 25, 2025
have you done DCX-Russian?
GoodMorning   82
N Apr 24, 2025 by EpicBird08
Source: 2023 USAJMO Problem 3
Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$.

Proposed by Holden Mui
82 replies
GoodMorning
Mar 23, 2023
EpicBird08
Apr 24, 2025
Concurrency
shobber   9
N May 6, 2024 by joshualiu315
Source: APMO 1992
In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$.

Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.
9 replies
shobber
Mar 11, 2006
joshualiu315
May 6, 2024
Concurrency
G H J
G H BBookmark kLocked kLocked NReply
Source: APMO 1992
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shobber
3498 posts
#1 • 2 Y
Y by Adventure10, Mango247
In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$.

Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.
Z K Y
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yetti
2643 posts
#2 • 2 Y
Y by Adventure10, Mango247
Assign weights $m_{A1} = OO_1 = r - r_1, m_O = O_1A_1 = r_1$ to $A_1, O$, respectively, so that these 2 weights balance at $O_1$, which is then assigned weight $m_{O1} = m_{A1} + m_O = r - r_1 + r_1 = r$. Assign weight $m_{O2}$ to $O_2$, such that the weights $m_{O1}, m_{O2}$ balance at $A$:

$\frac{m_{O2}}{m_{O1}} = \frac{O_1A}{O_2A} = \frac{r_1}{r_2}$

$m_{O2} = m_{O1}\ \frac{r_1}{r_2} = \frac{r r_1}{r_2}$

Assign weight $m_{A2}$ to $A_2$, so that the weights $m_O, m_{A2}$ balance at $O_2$:

$\frac{m_{A2}}{m_O} = \frac{OO_2}{O_2A_2} = \frac{r - r_2}{r_2}$

$m_{A2} + m_O = m_{O2} = \frac{r r_1}{r_2}$

Solving for $m_{A2}$,

$m_{A2} = m_O\ \frac{r - r_2}{r_2} = \left(\frac{r r_1}{r_2} - m_{A2}\right) \frac{r - r_2}{r_2}$

$m_{A2} \left(1 + \frac{r - r_2}{r_2}\right) = \frac{r r_1}{r_2} \cdot \frac{r - r_2}{r_2}$

$m_{A2} = \frac{r_1(r - r_2)}{r_2}$

Let $OA$ meet $A_1A_2$ at $P$. The weights $m_{A1}, m_{A2}$ now balance at $P$, i.e.,

$\frac{A_1P}{A_2P} = \frac{m_{A2}}{m_{A1}} = \frac{r_1(r - r_2)}{r_2(r - r_1)}$

Since we also have $\frac{OO_1}{A_1O_1} = \frac{r - r_1}{r_1},\ \frac{OO_2}{A_2O_2} = \frac{r - r_2}{r_2}$, it follows that

$\frac{OO_1}{O_1A_1} \cdot \frac{A_1P}{PA_2} \cdot \frac{A_2O_2}{O_2O} = \frac{r - r_1}{r_1} \cdot \frac{r_1(r - r_2)}{r_2(r - r_1)} \cdot \frac{r_2}{r - r_2} = 1$

By Ceva's theorem, the cevians $A_2O_1,\ OA \equiv OP,\ A_1O_2$ in the triangle $\triangle OA_1A_2$ are concurrent.
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neverstop
9 posts
#3 • 3 Y
Y by Adventure10, Mango247, Stuffybear
in this problem, we just apply the ceva theorem with triangle $OO_1O_2$, with A in $O_1O_2$, $A_1$ in $OO_1$, $A_2$ in $OO_2$
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pontios
777 posts
#4 • 2 Y
Y by Adventure10, Mango247
shobber wrote:
Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.

This result is also true if the three circles are externally tangent.

That's what neverstop means and this is true because $\frac{s-a}{s-b} \cdot \frac{s-b}{s-c} \cdot \frac{s-c}{s-a} =1$
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Vo Duc Dien
341 posts
#5 • 2 Y
Y by Adventure10, Mango247
http://www.cut-the-knot.org/wiki-math/index.php?n=MathematicalOlympiads.APMO1992Problem2

Vo Duc Dien
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erfan_Ashorion
102 posts
#6 • 1 Y
Y by Adventure10
oh nice problem!!
lemma 1:$O1,O2,A$ are on same line.
proof:we know that $\angle O2AO =90$ $  \angle O1AO =90$ so $O1,A,O2$ are on same line :)
lemma 2:$O,O2,A2$are on same line and $O,O1,A1$ are on same line.
proof:we know that $O2A2$ is perpendicular to line that tangent to circle$C$ and we know that $OA2$ is perpendicular to line that tangent to circle$C$ so $A2,O2,O$ are on same line :wink: !
proof of problem:
suppose $A2A1O2 = X$ and $O2A1O=Y$ and $OA2O1 = P$ AND $O1A2A1=Q$ we must proof:
$\frac{\sin X}{\sin Y} . \frac{\sin AOO1}{\sin AOO2} . \frac{\sin P}{\sin Q} =1$
we know that :
$\frac{\sin AOO1}{\sin AOO2} = \frac {r_1}{r_2} . \frac{OO2}{OO1}$
and we know that:
$\frac{\sin X}{\sin Y} . \frac{AA2}{A1O} = \frac {r_2}{OO2}$
and also it is right for $\frac{\sin P}{\sin Q}$and we know that $OA2=OA1$ so the problem is proof!
excuse me because im so lazy to right all of my solution but im sure that you know that my solution :wink:
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armpist
527 posts
#7 • 3 Y
Y by Adventure10, Mango247, and 1 other user
Dear MLs

It is simple.

The concurrency point is this case is, obviously,
the Lemoine point (symmedian point) of triangle AA1A2.


r1 and r2 are two red herrings, while r isn't.


This post is dedicated to both Mr. Referendum - G. Papandreou and
Mr. M.I.T. - Darij, not necessarily in this order.

Friendly,
M.T.
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sunken rock
4396 posts
#8 • 2 Y
Y by Adventure10, Mango247
@armpist: very fine observation, although not quite 'obvious'; I think it can bring new good questions about this configuration.

Best regards,
sunken rock
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armpist
527 posts
#9 • 2 Y
Y by Adventure10, Mango247
@sunken rock

maybe the very first candidate should be the Sangaku of Gumma, even though there
are plenty of solutions by now.
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=18266

It may lead to a new approach.

M.T.
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joshualiu315
2534 posts
#10
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Apply Ceva's theorem on $\triangle OO_1O_2$. The following statement is equivalent to the desired lines being concurrent:

\[\frac{OA_1}{A_1O_1} \cdot \frac{O_1A}{AO_2} \cdot \frac{O_2A_2}{A_2O} = 1.\]
Then, notice that we can write everything in terms of $r$, $r_1$ and $r_2$. Since

\[\frac{r}{r_1} \cdot \frac{r_1}{r_2} \cdot \frac{r_2}{r} = 1,\]
we are done. $\square$
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