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k a My Retirement & New Leadership at AoPS
rrusczyk   1571
N Mar 26, 2025 by SmartGroot
I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE
1571 replies
rrusczyk
Mar 24, 2025
SmartGroot
Mar 26, 2025
k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/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.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
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Introduction to Algebra A Self-Paced

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Calculus
Sunday, Mar 30 - Oct 5
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Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Sunday, Mar 23 - Jun 15
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WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
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Programming

Introduction to Programming with Python
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Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Mar 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
CGMO6: Airline companies and cities
v_Enhance   13
N 3 minutes ago by Marcus_Zhang
Source: 2012 China Girl's Mathematical Olympiad
There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$.
13 replies
+1 w
v_Enhance
Aug 13, 2012
Marcus_Zhang
3 minutes ago
nice problem
hanzo.ei   0
17 minutes ago
Source: I forgot
Let triangle $ABC$ be inscribed in the circumcircle $(O)$ and circumscribed about the incircle $(I)$, with $AB < AC$. The incircle $(I)$ touches the sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A line through $I$, perpendicular to $AI$, intersects $BC$, $CA$, and $AB$ at $X$, $Y$, and $Z$, respectively. The line $AI$ meets $(O)$ at $M$ (distinct from $A$). The circumcircle of triangle $AYZ$ intersects $(O)$ at $N$ (distinct from $A$). Let $P$ be the midpoint of the arc $BAC$ of $(O)$. The line $AI$ cuts segments $DF$ and $DE$ at $K$ and $L$, respectively, and the tangents to the circle $(DKL)$ at $K$ and $L$ intersect at $T$. Prove that $AT \perp BC$.
0 replies
hanzo.ei
17 minutes ago
0 replies
Find a given number of divisors of ab
proglote   9
N 31 minutes ago by zuat.e
Source: Brazil MO 2013, problem #2
Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$). Then Arnaldo tells the number of divisors of $ab$. Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo.
9 replies
proglote
Oct 24, 2013
zuat.e
31 minutes ago
2025 TST 22
EthanWYX2009   1
N 38 minutes ago by hukilau17
Source: 2025 TST 22
Let \( A \) be a set of 2025 positive real numbers. For a subset \( T \subseteq A \), define \( M_T \) as the median of \( T \) when all elements of \( T \) are arranged in increasing order, with the convention that \( M_\emptyset = 0 \). Define
\[
P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.
\]Find the smallest real number \( C \) such that for any set \( A \) of 2025 positive real numbers, the following inequality holds:
\[
P(A) - Q(A) \leq C \cdot \max(A),
\]where \(\max(A)\) denotes the largest element in \( A \).
1 reply
EthanWYX2009
3 hours ago
hukilau17
38 minutes ago
AMC 10/AIME Study Forum
PatTheKing806   68
N 3 hours ago by PatTheKing806
[center]

Me (PatTheKing806) and EaZ_Shadow have created a AMC 10/AIME Study Forum! Hopefully, this forum wont die quickly. To signup, do /join or \join.

Click here to join! (or do some pushups) :P

People should join this forum if they are wanting to do well on the AMC 10 next year, trying get into AIME, or loves math!
68 replies
PatTheKing806
Thursday at 11:34 PM
PatTheKing806
3 hours ago
Incircles
r00tsOfUnity   15
N 3 hours ago by Mathgloggers
Source: 2024 AIME I #8
Eight circles of radius $34$ can be placed tangent to side $\overline{BC}$ of $\triangle ABC$ such that the first circle is tangent to $\overline{AB}$, subsequent circles are externally tangent to each other, and the last is tangent to $\overline{AC}$. Similarly, $2024$ circles of radius $1$ can also be placed along $\overline{BC}$ in this manner. The inradius of $\triangle ABC$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
15 replies
r00tsOfUnity
Feb 2, 2024
Mathgloggers
3 hours ago
Tennessee Math Tournament (TMT) Online 2025
TennesseeMathTournament   54
N 5 hours ago by Inaaya
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 12th, 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!

Thank you to our lead sponsor, Jane Street!

IMAGE
54 replies
TennesseeMathTournament
Mar 9, 2025
Inaaya
5 hours ago
Pascal, Cayley and Fermat 2025
melpomene7   50
N 5 hours ago by Cerberusman
Anyone else do a CEMC contest? I did fermat but totally fumbled and got a 108.
50 replies
melpomene7
Feb 28, 2025
Cerberusman
5 hours ago
Subset coloring
v_Enhance   72
N Today at 8:02 AM by Mathgloggers
Source: USAMO 2015 Problem 3
Let $S = \left\{ 1,2,\dots,n \right\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of $T$ that are blue.

Determine the number of colorings that satisfy the following condition: for any subsets $T_1$ and $T_2$ of $S$, \[ f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2). \]
72 replies
v_Enhance
Apr 28, 2015
Mathgloggers
Today at 8:02 AM
[Registration Open] Gunn Math Competition is BACK!!!
the_math_prodigy   16
N Today at 5:13 AM by the_math_prodigy
Source: compete.gunnmathcircle.org
IMAGE

UPDATE! We now offer GMC online as hosted on MathDash! Visit our https://mathdash.com/channel/gmc-7vuxi for more info!

Gunn Math Competition will take place at Gunn High School in Palo Alto, California on THIS Sunday, March 30th. Gather a team of up to four and compete for over $7,500 in prizes! The deadline to sign up is March 27th. We welcome participants of all skill levels, with separate Beginner and Advanced (AIME) divisions for all students, from advanced 4th graders to 12th graders.

For more information, check our MathDash Channel, [url][/url]https://mathdash.com/channel/gmc-7vuxi, where registration is free and now open. The deadline to sign up is this Friday, March 28th. If you are unable to make a team, register as an individual and we will be able to create teams for you.

Special Guest Speaker: Po-Shen LohIMAGE
We are honored to welcome Po-Shen Loh, a world-renowned mathematician, Carnegie Mellon professor, and former coach of the USA International Math Olympiad team. He will deliver a several 30-minute talks to both students and parents, offering deep insights into mathematical thinking and problem-solving in the age of AI!

For any questions, reach out at ghsmathcircle@gmail.com or ask in our Discord server, which you can join through the website.

Find information on our AoPS page too! https://artofproblemsolving.com/wiki/index.php/Gunn_Math_Competition_(GMC)
Thank you to our sponsors for making this possible!
IMAGE

Check out our flyer! IMAGE
16 replies
the_math_prodigy
Mar 24, 2025
the_math_prodigy
Today at 5:13 AM
Why did my account set back to bronze?
Wisteriaaa   2
N Today at 3:10 AM by ethan2011
Hey, guys! I passed bronze division in Feb 2024 and silver in Jan 2025, but then I found I only could enter bronze division in Feb. What's going wrong with my account? :wacko:
2 replies
Wisteriaaa
Today at 3:02 AM
ethan2011
Today at 3:10 AM
USA Canada math camp
Bread10   39
N Today at 2:47 AM by Ilikeminecraft
How difficult is it to get into USA Canada math camp? What should be expected from an accepted applicant in terms of the qualifying quiz, essays and other awards or math context?
39 replies
Bread10
Mar 2, 2025
Ilikeminecraft
Today at 2:47 AM
2025 INTEGIRLS NYC/NJ Math Competition
sargamsujit   2
N Today at 12:02 AM by sargamsujit
NYC/NJ INTEGIRLS will be hosting our second annual math competition on May 3rd, 2025 from 9:30 AM to 4:30 PM EST at Rutgers University. Last year, we proudly organized the largest math competition for girls globally, welcoming over 500 participants from across the tristate area. Join other female-identifying and non-binary "STEMinists" in solving problems, socializing, playing games, and more! If you are interested in competing, please register at https://forms.gle/jqwEiq5PgqefetLj7

Find our website at https://nyc.nj.integirls.org/

[center]Important Information[/center]

Eligibility: This competition is open to all female-identifying and non-binary students in 8th grade or under. The competition is also completely free, including registration and lunch.

System: We will have two divisions: a middle school division and an elementary school division. There will be an individual round and team 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 elementary school problems will range from introductory to AMC 8 level, while the middle school problems will be for more advanced problem-solvers. Team round problems will cover various difficulty levels.

Platform: This contest will be held in person at Rutgers University. Competitors will all receive free merchandise, raffle tickets, and the chance to win exclusive gift prizes!


[center]Prizes

Over $2,000 in awards, including plaques, medals, plushies, gift cards, toys, books, swag, and more for top competitors and teams

[center]Help Us Out[/center]


[center]Please help us in sharing our competition and spreading the word! 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!
Format credits go to Indy INTEGIRLS!
2 replies
sargamsujit
Jan 28, 2025
sargamsujit
Today at 12:02 AM
A lot of integer lengths: JMO #6 or USAMO Problem 4
BarbieRocks   80
N Yesterday at 10:25 PM by Maximilian113
Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.
80 replies
BarbieRocks
Apr 29, 2010
Maximilian113
Yesterday at 10:25 PM
Inspired by IMO 1984
sqing   3
N Mar 26, 2025 by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+c+ ab +9abc\leq 1$$$$ a^2+b^2+c +ab+10 abc\leq\frac{28}{27}$$$$a^2+b^2+c+ ab +\frac{19}{2}abc\leq\frac{55}{54}$$
3 replies
sqing
Mar 26, 2025
sqing
Mar 26, 2025
Inspired by IMO 1984
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G H BBookmark kLocked kLocked NReply
Source: Own
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sqing
41310 posts
#1
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Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+c+ ab +9abc\leq 1$$$$ a^2+b^2+c +ab+10 abc\leq\frac{28}{27}$$$$a^2+b^2+c+ ab +\frac{19}{2}abc\leq\frac{55}{54}$$
This post has been edited 1 time. Last edited by sqing, Mar 26, 2025, 7:43 AM
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sqing
41310 posts
#2
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Let $ a,b,c\geq 0 $ and $  a^3+b^2+c+ ab +\frac{19}{2}abc\geq\frac{55}{54} $. Prove that
$$ a+b+c \geq  \frac{1}{3}\sqrt{\frac{55}{2}}$$$$ a+b+\frac{9}{10}c \geq \frac{11}{12}$$Let $ a,b,c\geq 0 $ and $  a^3+b^2+c +ab+10 abc\geq\frac{28}{27} $. Prove that
$$ a+b+c \geq  \frac{\sqrt[3]{28}}{3}$$$$ a+b+\frac{9}{10}c \geq \frac{14}{15}$$
This post has been edited 1 time. Last edited by sqing, Mar 26, 2025, 8:35 AM
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lbh_qys
450 posts
#3
Y by
sqing wrote:
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+c+ ab +9abc\leq 1$$$$ a^2+b^2+c +ab+10 abc\leq\frac{28}{27}$$$$a^2+b^2+c+ ab +\frac{19}{2}abc\leq\frac{55}{54}$$

(1).
\[a^2 + b^2 + c + ab + 9abc \leq a^2 + b^2 + c + ab + (ab+bc+ca) = 1\](2).(3).
This is merely based on (1) by employing \(abc \leq \frac{1}{27}\).
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sqing
41310 posts
#4
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Very nice.Thank lbh_qys.
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