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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
Find the angle alpha [Iran Second Round 1994]
Amir Hossein   4
N a few seconds ago by Mysteriouxxx
In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$
IMAGE
4 replies
Amir Hossein
Nov 26, 2010
Mysteriouxxx
a few seconds ago
Inequality
lgx57   1
N 6 minutes ago by DAVROS
Source: Own
$a,b>0$,$a^4+a^2b^2+b^4=k$.Find the min of $4a^2-ab+4b^2$.

$a,b>0$,$a^4-a^2b^2+b^4=k$.Find the min of $4a^2-ab+4b^2$.
1 reply
lgx57
5 hours ago
DAVROS
6 minutes ago
Calculating sum of the numbers
Sadigly   2
N 25 minutes ago by Gggvds1
Source: Azerbaijan Junior MO 2025 P4
A $3\times3$ square is filled with numbers $1;2;3...;9$.The numbers inside four $2\times2$ squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?

$\text{a)}$ $24,24,25,25$

$\text{b)}$ $20,23,26,29$
2 replies
Sadigly
6 hours ago
Gggvds1
25 minutes ago
Help me identify what should i focus in alcumus for contest's each problem
Hope_and_fight   0
28 minutes ago

So this file i have attached is a sample test from my upcoming regional schools' math contest. This is just a test sample, i was looking for problems as similar as to these so i have more material to practice with. Unfortunately i don't have much time to read the whole books the contest is already soon. I want to test myself as much as possible with problems, also Alcumus shows from what section of the book it is. so i can kinda cram the pages

TLDR: I WOULD BE REALLY GRATEFUL IF YOU COULD POINT ON WHAT TO FOCUS ON ALCUMUS FOR EACH OF THESE CONTEST PROBLEMS. AND SORRY FOR MY ENGLISH. I DON'T KNOW MUCH OF MATH IN THERE:)
0 replies
Hope_and_fight
28 minutes ago
0 replies
HCSSiM vs other programs
MathWizardThatCanBeatYou   2
N Today at 4:17 AM by MasterInTheMaking
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?
2 replies
MathWizardThatCanBeatYou
May 6, 2025
MasterInTheMaking
Today at 4:17 AM
9 ARML Location
deduck   38
N Today at 3:45 AM by shawnzeng
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
38 replies
deduck
May 6, 2025
shawnzeng
Today at 3:45 AM
USAMO Medals
YauYauFilter   54
N Today at 3:26 AM by ohiorizzler1434
YauYauFilter
Apr 24, 2025
ohiorizzler1434
Today at 3:26 AM
Will I fail again
hashbrown2009   16
N Today at 3:24 AM by ohiorizzler1434
so this year I got 34 on JMO 772 774 and got docked 1 point from top honors + mop

I just got info that I pretty much cannot do math for the rest of summer due to family reasons, and the only time I have is winter break

do you guys think it's enough time to practice/grind to qualify mop through USAMO, or should I tell my parents to reschedule the stuff because I really want to make mop

(Note: I'm aiming for like 25+ on USAMO so at least silver but I'm not sure that's realistic given the circumstances i'm in)
16 replies
hashbrown2009
Yesterday at 1:54 PM
ohiorizzler1434
Today at 3:24 AM
Wizard101
El_Ectric   65
N Today at 3:13 AM by HamstPan38825
Source: USAMO 2016, Problem 6
Integers $n$ and $k$ are given, with $n\ge k\ge2$. You play the following game against an evil wizard.

The wizard has $2n$ cards; for each $i=1,\ldots,n$, there are two cards labeled $i$. Initially, the wizard places all cards face down in a row, in unknown order.

You may repeatedly make moves of the following form: you point to any $k$ of the cards. The wizard then turns those cards face up. If any two of the cards match, the game is over and you win. Otherwise, you must look away, while the wizard arbitrarily permutes the $k$ chosen cards and then turns them back face-down. Then, it is your turn again.

We say this game is winnable if there exist some positive integer $m$ and some strategy that is guaranteed to win in at most $m$ moves, no matter how the wizard responds.

For which values of $n$ and $k$ is the game winnable?
65 replies
El_Ectric
Apr 20, 2016
HamstPan38825
Today at 3:13 AM
SUMaC Residential vs. Ross
AwesomeDude10   9
N Today at 2:43 AM by Rong0625
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!
9 replies
AwesomeDude10
May 6, 2025
Rong0625
Today at 2:43 AM
HCSSiM results
SurvivingInEnglish   63
N Yesterday at 11:52 PM by KevinChen_Yay
Anyone already got results for HCSSiM? Are there any point in sending additional work if I applied on March 19?
63 replies
SurvivingInEnglish
Apr 5, 2024
KevinChen_Yay
Yesterday at 11:52 PM
Sad Algebra
tastymath75025   46
N Yesterday at 11:40 PM by Ilikeminecraft
Source: 2019 USAMO 6, by Titu Andreescu and Gabriel Dospinescu
Find all polynomials $P$ with real coefficients such that $$\frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x)$$holds for all nonzero real numbers $x,y,z$ satisfying $2xyz=x+y+z$.

Proposed by Titu Andreescu and Gabriel Dospinescu
46 replies
tastymath75025
Apr 18, 2019
Ilikeminecraft
Yesterday at 11:40 PM
Aime ll 2022 problem 5
Rook567   1
N Yesterday at 9:39 PM by clarkculus
I don’t understand the solution. I got 220 as answer. Why does it insist, for example two primes must add to the third, when you can take 2,19,19 or 2,7,11 which for drawing purposes is equivalent to 1,1,2 and 2,7,9?
1 reply
Rook567
Yesterday at 9:08 PM
clarkculus
Yesterday at 9:39 PM
MathILy 2025 Decisions Thread
mysterynotfound   41
N Yesterday at 9:11 PM by cappucher
Discuss your decisions here!
also share any relevant details about your decisions if you want
41 replies
mysterynotfound
Apr 21, 2025
cappucher
Yesterday at 9:11 PM
question 6
nima1376   9
N Mar 10, 2025 by bin_sherlo
Source: iran tst 2014 third exam
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC$ at $D$.
let $X$ is a point on arc $BC$ from circumcircle of triangle $ABC$ such that if $E,F$ are feet of perpendicular from $X$ on $BI,CI$ and $M$ is midpoint of $EF$ we have $MB=MC$.
prove that $\widehat{BAD}=\widehat{CAX}$
9 replies
nima1376
May 21, 2014
bin_sherlo
Mar 10, 2025
question 6
G H J
Source: iran tst 2014 third exam
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nima1376
111 posts
#1 • 3 Y
Y by anantmudgal09, Adventure10, Mango247
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC$ at $D$.
let $X$ is a point on arc $BC$ from circumcircle of triangle $ABC$ such that if $E,F$ are feet of perpendicular from $X$ on $BI,CI$ and $M$ is midpoint of $EF$ we have $MB=MC$.
prove that $\widehat{BAD}=\widehat{CAX}$
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nima1376
111 posts
#2 • 8 Y
Y by anonymouslonely, wiseman, TheBeatlesVN, enzoP14, Chokechoke, Adventure10, Mango247, and 1 other user
solution:
let $\widehat{CBA}=2a,\widehat{BCA}=2b$ .

let $Q,W,R$ are foot of perpendicular from $E,M,F$ on $BC$.
$MB=MC$, $\Rightarrow$ $BW=WC$. but $M$ is mid point of $EF\Rightarrow WQ=WR \Rightarrow BQ=CR \Rightarrow BE.\cos a =CF.\cos b$ .

let $I_{a}$ is $A$-excenter.

let $T,S$ are foot of perpendicular from $X$ to $BI_{a},CI_{a}$, $\Rightarrow \frac{XT}{XS}=\frac{BE}{CF}=\frac{\cos b}{\cos a}=\frac{\sin \widehat{BCI_{a}}}{\sin \widehat{CBI_{a}}}=\frac{\sin \widehat{XI_{a}B}}{\sin \widehat{XI_{a}C}}\Rightarrow XI_{a}$ is symmedian of triangle $CI_{a}B$ .

let $L$ is midpoint of arc $BAC$ .$\widehat{LBC}=\widehat{LCB}=\widehat{CI_{a}}B\Rightarrow LB,LC$ are tangent to circumcircle of triangle $CI_{a}B$.
$\frac{BX}{CX}=\frac{\sin \widehat{BLX}}{\sin \widehat{CLX}}=(\frac{\cos b}{\cos a})^{2}$ $(1)$

$\frac{\sin \widehat{BAD}}{\sin \widehat{CAD}}=\frac{AC.BD}{AB.CD}=\frac{\tan b}{\tan a}.\frac{sin 2a}{\sin 2b}=(\frac{\cos a}{\cos b})^{2}.(2)$

$(1,2)\,\Rightarrow \frac{\sin \widehat{BAD}}{\sin \widehat{CAD}}=\frac{\sin \widehat{CLX}}{\sin \widehat{XLB}},(\widehat{BAD}+\widehat{CAD}=\widehat{CLX}+\widehat{BLX})\Rightarrow \widehat{CAX}=\widehat{CLX}=\widehat{BAD}$ .

so we are done.
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leader
339 posts
#3 • 2 Y
Y by wiseman, Adventure10
I have a long solution.
Begin torment
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IDMasterz
1412 posts
#4 • 2 Y
Y by wiseman, Adventure10
Note that it suffice to show $I_A, X, M_A$ are collinear, where $I_A, M_A$ are the A-excentre and midpoint of arc $BAC$ respectively.

I cant see a nicer way to show this without ratios... First, it is not too difficult to show for a point $X$ satisfying the property, $BE/CF$ is constant, which in this case is $I_AB/I_AC$. Immediately, using $M_AB = M_AC, II_A=II_A, BP = CQ$ where $P, Q$ are the feet of $E, F$ onto $BC$ and letting $Y, Z$ be the feet of $M_A$ on $BI, IC$, that $BE/BY = CE/CZ$. This means that $I_A, N, M_A$ are collinear by affinity.

To see why this implies the result, we use the extraversion that the A-mixtilinear incircle point, $I$, $M_A$ are collinear. Indeed, $X$ is just the A-mixtilinear excircle point.
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kapilpavase
595 posts
#5 • 7 Y
Y by anantmudgal09, Ankoganit, Aryan-23, HolyMath, rashah76, Adventure10, Mango247
The crux part of this problem is that the property of point $X$ to be proved actually holds for all points on line $I_AN$, where $N$ is the midpoint of arc $BAC$. This is what makes the problem hard, because there is not much special about the point $X$ which can enable one to find a neat synthetic proof without the fuss of ratio chasing things. But the above mentioned generalisation can be employed in the dynamic sense to excavate the roots of problem. Hence in my opinion, the approach below should completely demystify the problem.

First of all, as the above posts have noted, if $X$ is the intersection of $I_AN$ and $\odot ABC$, then $\angle{BAD}=\angle{CAX}$, which is the well known extraversion of mixtilinear touchpoint property.

In the proof, we refer to point $X$ as a variable point on line $I_AN$ , $E,F$ the perpendiculars from $X$ to $BI,CI$ and $M$ as the midpoint of $E,F$. Thus in this context, $E,F,M$ are variable points. We shall nail the problem in two steps. First we show that the locus of $M$ must be a straight line, then we show this line is the perpendicular bisector of $BC$ and we would be through.

STEP 1

STEP 2
This post has been edited 6 times. Last edited by kapilpavase, Dec 30, 2016, 7:54 AM
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anantmudgal09
1980 posts
#6 • 3 Y
Y by SerdarBozdag, Adventure10, Mango247
It is like the IMO 2015 G4 where a point which typically lies on a line is said to lie on a circle (meaning it's their intersection) and some property is asked to hold.

Solution
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nguyenhaan2209
111 posts
#7 • 2 Y
Y by top1csp2020, Adventure10
We have a lemma; X is the second intersection of A-mixtilinear excircle with (O) then AD, AX isogonal
$1$. $IaX$ is bisector of $BXC$: prove similarly to A-inmixtilinear (generalized by Prostasov)
$2$. $S$ is orthocentre of $IBC$ then $S,M,Ia$ collinear and it's easy to see that $AI/AIa=SI/SD$ so $IaM//AD$
$3$.$IAD=AIaM=RNIa(IaR^2=RM.RN)=RAX$ so $AD, AX$ isogonal
Back to the problem: From the lemma $BAD=CAX (1)$
By angle chasing $I$ is orthocentre of $EMF$ then $NEMF$ is a pallelogram so $GB=GC$
Apply $E.R.I.Q$ lemma to $(X,N,I_a)$ wrt $IB$,$IC$ because $EK/EC=NX/NIa=FB/FJ$ so we have $H$ lies on $GM$ hence $HB=HC (2)$
Assume that there exist $X'$ distinct from $X$ satisfy $(ii)$ apply $ERIQ$ lemma again it mean $J'B/K'C=FB/EC=JB/KC$ so $J', K'$ lies on the same plane with $JK$ so $X'$ not lies on $(O) (3)$
From $(1)(2)(3)$ so Q.E.D
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yayups
1614 posts
#8 • 2 Y
Y by Gaussian_cyber, Adventure10
This is evidently a straightforward complex bash. Relabel $X$ in the problem to $T$. Let $(ABC)$ be the unit circle with $a=x^2$, $b=y^2$, $c=z^2$, such that the midpoints of the arcs are $p=-yz$, $q=-zx$, $r=-xy$. Note that $E$ is the foot from $T$ to $BQ$, so \[e=\frac{1}{2}(t-xz+y^2+xzy^2/t).\]We similarly derive \[f=\frac{1}{2}(t-xy+z^2+xyz^2/t),\]so \[m=\frac{t}{2}+\frac{y^2+z^2}{4}+\frac{(y+z)x}{4}(yz/t-1).\]The problem tells us that $M$ is on the perpendicular bisector of $BC$, which in our case means that $m/(y^2+z^2)$ is real, or that \[k:=\frac{m}{y^2+z^2}-\frac{1}{4}=\frac{2t+\frac{x(y+z)}{t}(yz-t)}{y^2+z^2}\]is real. This means that \[\frac{2t+\frac{x(y+z)}{t}(yz-t)}{y^2+z^2}=\frac{\frac{2}{t}+\frac{t(y+z)}{xyz}\frac{t-yz}{yzt}}{\frac{y^2+z^2}{y^2z^2}},\]or \[2\left(t-\frac{y^2z^2}{t}\right)=(y+z)(t-yz)\left(\frac{1}{x}+\frac{x}{t}\right).\]Note that $t\ne yz$, as that is not on arc $BC$, so we may cancel the factor $t-yz$ from both sides to obtain the equation \[2\frac{t+yz}{t}=\frac{y+z}{tx}(t+x^2),\]so $t=\boxed{\frac{x(2yz-xy-xz)}{y+z-2x}}$.

It suffices to show that $T$ is the mixtilinear excircle touch point. To do this, we use the fact that $T,I_A,L$ are collinear, where $\ell=yz$ is the arc midpoint of $BAC$. We'll solve for $t$ using this fact, and show that we get the same formula. Note that $I_A=xy+xz-yz$, and we have $I_A$ on the chord $TL$, so \[I_A+\ell t\bar{I}_A=\ell+t,\]or \[t=\frac{I_A-\ell}{1-\ell\bar{I}_A}=\frac{xy+xz-2yz}{1-\frac{1}{x}(z+y-x)}=\frac{x(2yz-xy-xz)}{y+z-2x},\]so we're done.

Remark: This complex bash actually motivates a short synthetic solution (which I think is really hard to motivate without this). Note that our proof showed that $L$ also satisfied the condition, since we factored out $t-yz=t-\ell$. Now, the value of the midpoint of $EF$ is linear in $T$, so since $L$ and the mixtilinear extouch point work, any point on that line must also work. From this, we could have originally just verified the problem for $I_A$ and $L$, and finished that way, but again, I don't see how one can come up with this.
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nbdaaa
347 posts
#9
Y by
I just want to clarify a claim used by some of the above posts that doesn't seem to be trivial to me :(

Claim: Given $\triangle ABC$ with incenter $I$, $I_a,I_b,I_c$ be the A-excenter, B-excenter, C-excenter of $\triangle ABC$. Let $N$ be the midpoint of arc $BAC$. Let $X$ be an arbitrary point moving on $NI_a$. $E,F$ be the projection of $X$ on $BI,CI$. Then prove that the locus of the midpoint $P$ of $EF$ is the perpendicular bisector of $BC$

Proof
Let $S$ be the projection of $I$ on $NI_a$
We have $\overrightarrow {PQ}  = \overrightarrow {SQ}  - \overrightarrow {SP}  = \frac{{\overrightarrow {SB}  + \overrightarrow {SC} }}{2} - \frac{{\overrightarrow {SE}  + \overrightarrow {SF} }}{2} = \frac{{(\overrightarrow {SB}  - \overrightarrow {SE} ) + (\overrightarrow {SC}  - \overrightarrow {SF} )}}{2} = \frac{{\overrightarrow {EB}  + \overrightarrow {FC} }}{2}$
So we have to prove that \[\begin{array}{l}
PQ \perp BC \Leftrightarrow \overrightarrow {PQ} .\overrightarrow {BC}  = 0 \Leftrightarrow (\overrightarrow {BE}  + \overrightarrow {CF} ).\overrightarrow {BC}  = 0 \Leftrightarrow \overrightarrow {BE} .\overrightarrow {BC}  - \overrightarrow {FC} .\overrightarrow {BC}  = 0\\
\\
 \Leftrightarrow BE.BC.\cos \frac{B}{2} - FC.BC.\cos \frac{C}{2} = 0 \Leftrightarrow \frac{{BE}}{{CF}} = \frac{{\cos \frac{C}{2}}}{{\cos \frac{B}{2}}}
\end{array}\]Notice that there exists a spiral similarity with center $S$ sends $E$ to $B$, sends $F$ to $C$ and sends $P$ to the midpoint $Q$ of $BC$, then \[\frac{{BE}}{{CF}} = \frac{{SB}}{{SC}}\]Finally by applying the law of sine, we have \[\frac{{SB}}{{SC}} = \frac{{\sin SCB}}{{\sin SBC}} = \frac{{\sin N{I_a}B}}{{\sin A{I_c}B}} = \frac{{\cos \frac{C}{2}}}{{\cos \frac{B}{2}}}\]
This post has been edited 3 times. Last edited by nbdaaa, May 28, 2022, 3:10 PM
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bin_sherlo
720 posts
#10
Y by
Let $J$ be $A-$excenter, $P$ be the midpoint of arc $BC$. We want to show that $X$ is $A-$exmixtilinear touch point hence it sufficies to show that $X$ is $J-$dumpty on $\triangle JBC$.

Claim: $JBE\overset{-}{\sim} JCF$.
Proof: Let $BF\cap CE=G$. Since $PM$ is the newton-gauss line of quadrilateral $BCFE$, it passes through the midpoint of $IG$. This gives $JG\perp BC$. DDIT on quadrilateral $IGEF$ gives $(\overline{JI},\overline{JG}),(\overline{JE},\overline{JF}),(\overline{JB},\overline{JC})$ is an involution. This must be reflection over the angle bisector of $\measuredangle BJC$ hence $\measuredangle BJE=\measuredangle FJC$. Also $\measuredangle EBJ=90=\measuredangle JCF$ thus, the claim follows.

Claim: $X$ is $J-$dumpty on $\triangle JBC$.
Proof: Let $X'$ be $J-$dumpty.
\[\frac{\sin X'JC}{\sin X'JB}=\frac{JC}{JB}=\frac{CF}{BE}=\frac{\frac{CF}{JX}}{\frac{BE}{JX}}=\frac{\sin XJC}{\sin XJB}\]Since $\measuredangle BJX'+\measuredangle X'JC=\measuredangle BJX+\measuredangle XJC$ we get that $X=X'$ as desired.$\blacksquare$
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