Problem. Given non-negative real numbers satisfying Prove that Proposed by Phan Ngoc Chau

Let Prove that

Mustang Math is excited to announce that registration for our annual tournament, MMT 2025, is open! This year, we are bringing our tournament to 9 in-person locations, as well as online!

Locations include: Colorado, Norcal, Socal, Georgia, Illinois, Massachusetts, New Jersey, Nevada, Washington, and online. For registration and more information, check out https://mustangmath.com/competitions/mmt-2025.

MMT 2025 is a math tournament run by a group of 150+ mathematically experienced high school and college students who are dedicated to providing a high-quality and enjoyable contest for middle school students. Our tournament centers around teamwork and collaboration, incentivizing students to work with their teams not only to navigate the challenging and interesting problems of the tournament but also to develop strategies to master the unique rounds. This includes a logic puzzle round, a strategy-filled hexes round, a race-like gallop round, and our trademark ‘Mystery Mare’ round!

Awards:
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[*] Medals for the top teams
[*] Shirts, pins, stickers and certificates for all participants
[*] Additional awards provided by our wonderful sponsors!
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We are also holding a free MMT prep seminar from 3/15-3/16 to help students prepare for the upcoming tournament. Join the Google Classroom! https://classroom.google.com/c/NzQ5NDUyNDY2NjM1?cjc=7sogth4

Hi everyone,

Did anyone get their JSMCR decisions back yet? They were supposed to release on 2/28

Thanks!

I'm in 8th grade and have never taken the AMC 10. I am currently in alg2. I have scored 20 on AMC 8 this year and 34 on the chapter math counts last year. Can I qualify for AIME. Also what should I practice AMC 10 next year?

Hello everyone! We are excited to announce a new competition, the Tennessee Math Tournament, created by the Tennessee Math Coalition! Anyone can participate in the virtual competition for free.

The testing window is from March 22nd to April 5th, 2025. Virtual competitors may participate in the competition at any time during that window.

The virtual competition consists of three rounds: Individual, Bullet, and Team. The Individual Round is 60 minutes long and consists of 30 questions (AMC 10 level). The Bullet Round is 20 minutes long and consists of 80 questions (Mathcounts Chapter level). The Team Round is 30 minutes long and consists of 16 questions (AMC 12 level). Virtual competitors may compete in teams of four, or choose to not participate in the team round.

To register and see more information, click here!

If you have any questions, please email connect@tnmathcoalition.org or reply to this thread!

do they send the Interesting Test to everyone who applied or do they read the friendly letter first and only send to the kids they like?

Isosceles triangle , with , is inscribed in circle . Let be an arbitrary point inside such that . Ray intersects again at (other than ). Point (other than ) is chosen on such that . Line intersects rays and at points and , respectively. Prove that .

Proposed by Anton Trygub

The International Math Competition (IMC) is essentially the elementary and middle school equivalent of the IMO, with individual and team rounds featuring both short-answer and proof-based problems. See past problems here.

Team USA is looking for 6th graders and below with AIME qualification or AMC 8 DHR (or equivalent), and for 9th graders and below with JMO or Mathcounts Nationals qualification. If you think you meet said criteria, fill out the initial form here.

Here are a couple quick links for further information:
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[*] Dr. Tao Hong's website, which contains a detailed recap of the 2024 competition (and previous years'), as well as Team USA's historical results. (You may recognize a couple names... @channing421 @vrondoS et al.: back me up here :P)
[*] My journal, which gives an insider's perspective on the camp :ninja:
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Satisfactory. Keep cooking.
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Problems are posted at https://web.evanchen.cc/mockaime.html#current now!

Like last year, we're running the OTIS Mock AIME 2025 again, except this time there will actually be both a I and a II because we had enough problems to pull it off. However, the two versions will feel quite different from each other:

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[*] The OTIS Mock AIME I is going to be tough. It will definitely be harder than the actual AIME, by perhaps 2 to 4 problems. But more tangibly, it will also have significant artistic license. Problems will freely assume IMO-style background throughout the test, and intentionally stretch the boundary of what constitutes an “AIME problem”.
[*] The OTIS Mock AIME II is meant to be more practically useful. It will adhere more closely to the difficulty and style of the real AIME. There will inevitably still be some more IMO-flavored problems, but they’ll appear later in the ordering.
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Like last time, all 30 problems are set by current and past OTIS students.

Details are written out at https://web.evanchen.cc/mockaime.html, but to highlight important info:
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[*] Free, obviously. Anyone can participate.
[*]Both tests will release sometime Dec 19th. You can do either/both.
[*]If you'd like to submit for scoring, you should do so by January 20th at 23:59 Pacific time (same deadline for both). Please hold off on public spoilers before then.
[*]Solutions, statistics, and maybe some high scores will be published shortly after that.
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Feel free to post questions, hype comments, etc. in this thread.

Let be a cyclic quadrilateral with and . Point and are selected on segment such that . Points and are selected on segment such that . Prove that is a cyclic quadrilateral.

Proposed by Evan O'Dorney

Just curious, what does everyone think about the prestige of a USAMO qualification vs a USAJMO qual? Obv USAMO > USAJMO but how much?

And while we’re at it, what about amc 10 dhr vs amc 12 dhr? Thoughts?

Edit: also how much is a perfect score on 10/12 worth? Asking for a friend….