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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this 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 a transformative 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][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/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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Visit the pages linked for full schedule details for each of these programs!


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0 replies
jlacosta
Jun 2, 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
Inspired by SunnyEvan
sqing   1
N 2 minutes ago by sqing
Source: Own
Let $ x,y \geq 0 ,  \frac{x^2}{x^3+y}+\frac{y^2}{x+y^3} \geq 1 .$ Prove that
$$ (x-\frac{1}{2})^2+(y+\frac{1}{2})^2 \leq \frac{5}{2} $$$$ (x-1)^2+(y+1)^2 \leq 5 $$$$ (x-2)^2+(y+2)^2 \leq 13$$$$ (x-\frac{1}{2})^2+(y+1)^2 \leq \frac{17}{4} $$$$ (x-1)^2+(y+2)^2 \leq 10  $$$$ (x-\frac{1}{2})^2+(y+2)^2 \leq \frac{37}{4}  $$
1 reply
1 viewing
sqing
Today at 5:18 AM
sqing
2 minutes ago
tangent circles
m4thbl3nd3r   1
N 18 minutes ago by Lil_flip38
Let $O,H,T$ be circumcenter, orthocenter and A-HM point of triangle $ABC$. Let $AH,AT$ intersect $(O)$ at $K,N$, respectively. Let $XYZ$ be the triangle formed by $TH,BC,KN$. Prove that $(XYZ)$ is tangent to $(O).$
1 reply
m4thbl3nd3r
2 hours ago
Lil_flip38
18 minutes ago
inequality
SunnyEvan   2
N 25 minutes ago by sqing
Let $ a,b > 0 ,$ such that : $ a+b \geq \frac{3(a^4+b^4)}{a^2+b^2+1}\sqrt{\frac{\frac{1}{a}+\frac{1}{b}}{a+b}}.$
Prove that: $$ \frac{3(a^2+b^2)+4}{a^4+b^4} \geq 3(a+b)-1 $$
2 replies
SunnyEvan
Today at 6:53 AM
sqing
25 minutes ago
inequality
SunnyEvan   8
N 29 minutes ago by sqing
Let $ x,y \geq 0 ,$ such that : $ \frac{x^2}{x^3+y}+\frac{y^2}{x+y^3} \geq 1 .$
Prove that : $$ x^2+y^2-xy \leq x+y $$$$ (x+\frac{1}{2})^2+(x+\frac{1}{2})^2 \leq \frac{5}{2} $$$$ (x+1)^2+(y+1)^2 \leq 5 $$$$ (x+2)^2+(y+2)^2 \leq 13 $$
8 replies
SunnyEvan
Yesterday at 1:51 PM
sqing
29 minutes ago
AMC 10/12 trainer
grapecoder   7
N Today at 2:47 AM by nxchman
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://amc.grapecoder.repl.co
And here's the code (if anyone's interested): https://github.com/megagames-me/amc-trainer

Any feedback/suggestions are appreciated!
7 replies
grapecoder
Oct 22, 2023
nxchman
Today at 2:47 AM
[$26K+ in PRIZES AND IVY LEAGUE MENTORSHIP] AASF Youth Ambassador for Science
petfoo   6
N Today at 2:47 AM by NoSignOfTheta
Hey everyone! Just made this post to share something that meant a lot to me last summer.

I participated in the Youth Ambassador for Science competition hosted by the Asian American Scholar Forum. It’s open to high school students (14–18), and it’s super simple (and free) to enter:

You make a short video about one of five Asian American science pioneers, post it on social media with their hashtag (#aasf_contest), and fill out a quick submission form.

What I got out of it last year:
[list][*] I won a $500 VISA gift card
[*] Got invited to the Asian American Pioneer Medal Symposium at Stanford University
[*]Met Fortune 500 CEOs (Founder of Zoom), professors from top schools (many from Princeton and Stanford), and even a Nobel Prize winner (current professor at Berkeley)
[*]And received 1-on-1 mentoring sessions with a Princeton professor[/list]

It was honestly one of the highlights of my summer. This year, I’m helping spread the word so more students can benefit.

If you’re into science, public speaking, social impact, or just want to connect with some inspiring people, I 100% recommend going for it.

:!: Deadline to apply: June 30, 2025
:arrow: More Info: aasforum.org/video-competition

p.s. i'm a rising freshman at one of the hypsm schools, so i'm sure the contest helped there as well :) have fun!
6 replies
petfoo
Wednesday at 3:27 PM
NoSignOfTheta
Today at 2:47 AM
AMC 10 Registry
Crimzion   5
N Today at 1:59 AM by nxchman
Just wondering how do i apply for amc 10 this year, maa website says info about last years.
5 replies
Crimzion
Yesterday at 3:55 AM
nxchman
Today at 1:59 AM
On a^4+b^4=c^4+d^4=e^5
v_Enhance   120
N Today at 1:50 AM by shendrew7
Source: USAMO 2015, Problem 5
Let $a$, $b$, $c$, $d$, $e$ be distinct positive integers such that $a^4+b^4=c^4+d^4=e^5$. Show that $ac+bd$ is a composite number.
120 replies
v_Enhance
Apr 29, 2015
shendrew7
Today at 1:50 AM
Mustang Math Recruitment is Open!
MustangMathTournament   25
N Today at 1:00 AM by KevinChen_Yay
The Interest Form for joining Mustang Math is open!

Hello all!

We're Mustang Math, and we are currently recruiting for the 2025-2026 year! If you are a high school or college student and are passionate about promoting an interest in competition math to younger students, you should strongly consider filling out the following form: https://link.mustangmath.com/join. Every member in MM truly has the potential to make a huge impact, no matter your experience!

About Mustang Math

Mustang Math is a nonprofit organization of high school and college volunteers that is dedicated to providing middle schoolers access to challenging, interesting, fun, and collaborative math competitions and resources. Having reached over 4000 U.S. competitors and 1150 international competitors in our first six years, we are excited to expand our team to offer our events to even more mathematically inclined students.

PROJECTS
We have worked on various math-related projects. Our annual team math competition, Mustang Math Tournament (MMT) recently ran. We hosted 8 in-person competitions based in Washington, NorCal, SoCal, Illinois, Georgia, Massachusetts, Nevada and New Jersey, as well as an online competition run nationally. In total, we had almost 900 competitors, and the students had glowing reviews of the event. MMT International will once again be running later in August, and with it, we anticipate our contest to reach over a thousand students.

In our classes, we teach students math in fun and engaging math lessons and help them discover the beauty of mathematics. Our aspiring tech team is working on a variety of unique projects like our website and custom test platform. We also have a newsletter, which, combined with our social media presence, helps to keep the mathematics community engaged with cool puzzles, tidbits, and information about the math world! Our design team ensures all our merch and material is aesthetically pleasing.

Some highlights of this past year include 1000+ students in our classes, AMC10 mock with 150+ participants, our monthly newsletter to a subscriber base of 6000+, creating 8 designs for 800 pieces of physical merchandise, as well as improving our custom website (mustangmath.com, 20k visits) and test-taking platform (comp.mt, 6500+ users).

Why Join Mustang Math?

As a non-profit organization on the rise, there are numerous opportunities for volunteers to share ideas and suggest projects that they are interested in. Through our organizational structure, members who are committed have the opportunity to become a part of the leadership team. Overall, working in the Mustang Math team is both a fun and fulfilling experience where volunteers are able to pursue their passion all while learning how to take initiative and work with peers. We welcome everyone interested in joining!

More Information

To learn more, visit https://link.mustangmath.com/RecruitmentInfo. If you have any questions or concerns, please email us at contact@mustangmath.com.

https://link.mustangmath.com/join
25 replies
MustangMathTournament
May 24, 2025
KevinChen_Yay
Today at 1:00 AM
[CASH PRIZES] IndyINTEGIRLS Spring Math Competition
Indy_Integirls   77
N Yesterday at 5:36 PM by OGMATH
[center]IMAGE

Greetings, AoPS! IndyINTEGIRLS will be hosting a virtual math competition on May 25,
2024 from 12 PM to 3 PM EST.
Join other woman-identifying and/or non-binary "STEMinists" in solving problems, socializing, playing games, winning prizes, and more! If you are interested in competing, please register here![/center]

----------

[center]Important Information[/center]

Eligibility: This competition is open to all woman-identifying and non-binary students in middle and high school. Non-Indiana residents and international students are welcome as well!

Format: There will be a middle school and high school division. In each separate division, there will be an individual round and a team round, where students are grouped into teams of 3-4 and collaboratively solve a set of difficult problems. There will also be a buzzer/countdown/Kahoot-style round, where students from both divisions are grouped together to compete in a MATHCOUNTS-style countdown round! There will be prizes for the top competitors in each division.

Problem Difficulty: Our amazing team of problem writers is working hard to ensure that there will be problems for problem-solvers of all levels! The middle school problems will range from MATHCOUNTS school round to AMC 10 level, while the high school problems will be for more advanced problem-solvers. The team round problems will cover various difficulty levels and are meant to be more difficult, while the countdown/buzzer/Kahoot round questions will be similar to MATHCOUNTS state to MATHCOUNTS Nationals countdown round in difficulty.

Platform: This contest will be held virtually through Zoom. All competitors are required to have their cameras turned on at all times unless they have a reason for otherwise. Proctors and volunteers will be monitoring students at all times to prevent cheating and to create a fair environment for all students.

Prizes: At this moment, prizes are TBD, and more information will be provided and attached to this post as the competition date approaches. Rest assured, IndyINTEGIRLS has historically given out very generous cash prizes, and we intend on maintaining this generosity into our Spring Competition.

Contact & Connect With Us: Email us at indy@integirls.org.

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[center]Help Us Out

Please help us in sharing the news of this competition! Our amazing team of officers has worked very hard to provide this educational opportunity to as many students as possible, and we would appreciate it if you could help us spread the word!
77 replies
Indy_Integirls
May 11, 2025
OGMATH
Yesterday at 5:36 PM
Frustration with Olympiad Geo
gulab_jamun   19
N Yesterday at 1:13 PM by Alex-131
Ok, so right now, I am doing the EGMO book by Evan Chen, but when it comes to problems, there are some that just genuinely frustrate me and I don't know how to deal with them. For example, I've spent 1.5 hrs on the second to last question in chapter 2, and used all the hints, and I still am stuck. It just frustrates me incredibly. Any tips on managing this? (or.... am I js crashing out too much?)
19 replies
gulab_jamun
May 29, 2025
Alex-131
Yesterday at 1:13 PM
MOP Emails Out! (2025)
Mathandski   122
N Yesterday at 9:24 AM by ohiorizzler1434
What an emotional roller coaster the past 34 days have been.

Congrats to all that qualified!
122 replies
Mathandski
Apr 22, 2025
ohiorizzler1434
Yesterday at 9:24 AM
How many approaches you got? (A lot)
IAmTheHazard   87
N Yesterday at 8:04 AM by MuradSafarli
Source: USAMO 2023/2
Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that, for all $x,y \in \mathbb{R}^+$,
$$f(xy+f(x))=xf(y)+2.$$
87 replies
IAmTheHazard
Mar 23, 2023
MuradSafarli
Yesterday at 8:04 AM
star on a quilt
derekwang2048   22
N Yesterday at 4:11 AM by RedFireTruck
Source: 2025 AMC 8 #1
The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire $4$-by-$4$ grid is covered by the star?

$\textbf{(A)}\ 40\qquad \textbf{(B)}\ 50\qquad \textbf{(C)}\ 60\qquad \textbf{(D)}\ 75\qquad \textbf{(E)}\ 80$
IMAGE

Thank you @zhenghua for the diagram!
22 replies
derekwang2048
Jan 30, 2025
RedFireTruck
Yesterday at 4:11 AM
Tangent circles
SinaQane   1
N Aug 28, 2021 by VicKmath7
Source: 239 2019 S5
Circle $\Gamma$ touches the circumcircle of triangle $ABC$ at point $R$, and it touches the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. Rays $PQ$ and $BC$ intersect at point $X$. The tangent line at point $R$ to the circle $\Gamma$ meets the segment $QX$ at point $Y$. The line segment $AX$ intersects the circumcircle of triangle $APQ$ at point $Z$. Prove that the circumscribed circles of triangles $ABC$ and $XY Z$ are tangent.
1 reply
SinaQane
Jul 31, 2020
VicKmath7
Aug 28, 2021
Tangent circles
G H J
G H BBookmark kLocked kLocked NReply
Source: 239 2019 S5
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SinaQane
198 posts
#1
Y by
Circle $\Gamma$ touches the circumcircle of triangle $ABC$ at point $R$, and it touches the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. Rays $PQ$ and $BC$ intersect at point $X$. The tangent line at point $R$ to the circle $\Gamma$ meets the segment $QX$ at point $Y$. The line segment $AX$ intersects the circumcircle of triangle $APQ$ at point $Z$. Prove that the circumscribed circles of triangles $ABC$ and $XY Z$ are tangent.
Z K Y
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VicKmath7
1391 posts
#2
Y by
Great problem! A sketch: The tangency point $T$ is the intersection of $(APQ)$ and $(ABC)$. By radical axis, $T$ is on $AY$. The hardest thing to notice is that $TX$ is external bisector of $<BTC$, provable by similarity. The rest is angle chase. Btw, this doesn't require to know anything about mixtilinear incircles.
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