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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Today at 3:18 PM
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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 events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/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

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Programming

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0 replies
1 viewing
jlacosta
Today at 3:18 PM
0 replies
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 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
Famous geo configuration appears on the district MO
AndreiVila   5
N 17 minutes ago by chirita.andrei
Source: Romanian District Olympiad 2025 10.4
Let $ABCDEF$ be a convex hexagon with $\angle A = \angle C=\angle E$ and $\angle B = \angle D=\angle F$.
[list=a]
[*] Prove that there is a unique point $P$ which is equidistant from sides $AB,CD$ and $EF$.
[*] If $G_1$ and $G_2$ are the centers of mass of $\triangle ACE$ and $\triangle BDF$, show that $\angle G_1PG_2=60^{\circ}$.
5 replies
2 viewing
AndreiVila
Mar 8, 2025
chirita.andrei
17 minutes ago
Game on a row of 9 squares
EmersonSoriano   0
18 minutes ago
Source: 2018 Peru TST Cono Sur P10
Let $n$ be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:

$(i)\:$ Choose a number of the form $2^j$, with $j$ a non-negative integer, and place it in an empty square.

$(ii)\:$ Choose two (not necessarily consecutive) squares containing the same number, say $2^j$. Replace the number in one of the squares with $2^{j+1}$ and erase the number in the other square.

At the end of the game, one square contains the number $2^n$, while the other squares are empty. Determine, as a function of $n$, the maximum number of turns Alex can make.
0 replies
EmersonSoriano
18 minutes ago
0 replies
Classic complex number geo
Ciobi_   1
N 21 minutes ago by TestX01
Source: Romania NMO 2025 10.1
Let $M$ be a point in the plane, distinct from the vertices of $\triangle ABC$. Consider $N,P,Q$ the reflections of $M$ with respect to lines $AB, BC$ and $CA$, in this order.
a) Prove that $N, P ,Q$ are collinear if and only if $M$ lies on the circumcircle of $\triangle ABC$.
b) If $M$ does not lie on the circumcircle of $\triangle ABC$ and the centroids of triangles $\triangle ABC$ and $\triangle NPQ$ coincide, prove that $\triangle ABC$ is equilateral.
1 reply
Ciobi_
Today at 12:56 PM
TestX01
21 minutes ago
The greatest length of a sequence that satisfies a special condition
EmersonSoriano   0
24 minutes ago
Source: 2018 Peru TST Cono Sur P9
Find the largest possible value of the positive integer $N$ given that there exist positive integers $a_1, a_2, \dots, a_N$ satisfying
$$ a_n = \sqrt{(a_{n-1})^2 + 2018 \, a_{n-2}}\:, \quad \text{for } n = 3,4,\dots,N. $$
0 replies
EmersonSoriano
24 minutes ago
0 replies
Isosceles Triangulation
worthawholebean   70
N 2 hours ago by akliu
Source: USAMO 2008 Problem 4
Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n - 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $ \mathcal{P}$ into $ n - 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.
70 replies
worthawholebean
May 1, 2008
akliu
2 hours ago
Westford Academy to host Middle School Math Competition
cyou   5
N 3 hours ago by Inaaya
Hi AOPS community,

We are excited to announce that Westford Academy (located in Westford, MA) will be hosting its first ever math competition for middle school students (grades 5-8).

Based in Massachusetts, this tournament hosts ambitious and mathematically skilled students in grades 5–8 to compete against other middle school math teams while fostering their problem-solving skills and preparing them to continue enriching their STEM skills in high school and in the future.

This competition will be held on April 12, 2025 from 12:00 PM to 5:00 PM and will feature 3 rounds (team, speed, and accuracy). The problems will be of similar difficulty for AMC 8-10 and were written by USA(J)MO and AIME qualifiers.

If you are in the Massachusetts area and are curious about Mathematics, we cordially invite you to sign up by scanning the QR code on the attached flyer. Please note that teams consist of 4-6 competitors, but if you prefer to register as an individual competitor, you will be randomly placed on a team of other individual competitors. Feel free to refer the attached flyer and website as needed.


https://sites.google.com/westfordk12.us/wamt/home?authuser=2
5 replies
cyou
Mar 25, 2025
Inaaya
3 hours ago
Permutations Part 1: 2010 USAJMO #1
tenniskidperson3   69
N 3 hours ago by akliu
A permutation of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.
69 replies
tenniskidperson3
Apr 29, 2010
akliu
3 hours ago
2025 USAMO Rubric
plang2008   17
N 5 hours ago by Mathandski
1. Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.

Rubric for Problem 1

2. Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0,\,a_1,\,\ldots,\,a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.

Rubric for Problem 2

3. Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds:
[center]For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such[/center]
[center]that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$.[/center]
Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.

Note: $\triangle UVW$ is directly similar to $\triangle XYZ$ if there exists a sequence of rotations, translations, and dilations sending $U$ to $X$, $V$ to $Y$, and $W$ to $Z$.

Rubric for Problem 3

4. Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

Rubric for Problem 4

5. Determine, with proof, all positive integers $k$ such that \[\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k\]is an integer for every positive integer $n$.

Rubric for Problem 5

6. Let $m$ and $n$ be positive integers with $m\geq n$. There are $m$ cupcakes of different flavors arranged around a circle and $n$ people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person $P$, it is possible to partition the circle of $m$ cupcakes into $n$ groups of consecutive cupcakes so that the sum of $P$'s scores of the cupcakes in each group is at least $1$. Prove that it is possible to distribute the $m$ cupcakes to the $n$ people so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.

Rubric for Problem 6
17 replies
plang2008
Today at 1:33 AM
Mathandski
5 hours ago
New Competition Series: The Million!
Mathdreams   6
N Today at 4:21 PM by KevinChen_Yay
Hello AOPS Community,

Recently, me and my friend compiled a set of the most high quality problems from our imagination into a problem set called the Million. This series has three contests, called the whun, thousand and Million respectively.

Unfortunately, it did not get the love it deserved on the OTIS discord. Hence, we post it here to share these problems with the AOPS community and hopefully allow all of you to enjoy these very interesting problems.

Good luck! Lastly, remember that MILLION ORZ!

Edit: We have just been informed this will be the Orange MOP series. Please pay close attention to these problems!
6 replies
Mathdreams
Today at 3:57 AM
KevinChen_Yay
Today at 4:21 PM
LMT Spring 2025 and Girls&#039; LMT 2025
vrondoS   20
N Today at 4:17 PM by ehe
The Lexington High School Math Team is proud to announce LMT Spring 2025 and our inaugural Girls’ LMT 2025! LMT is a competition for middle school students interested in math. Students can participate individually, or on teams of 4-6 members. This announcement contains information for BOTH competitions.

LMT Spring 2025 will take place from 8:30 AM-5:00 PM on Saturday, May 3rd at Lexington High School, 251 Waltham St., Lexington, MA 02421.

The competition will include two individual rounds, a Team Round, and a Guts Round, with a break for lunch and mini-events. A detailed schedule is available at https://lhsmath.org/LMT/Schedule.

There is a $15 fee per participant, paid on the day of the competition. Pizza will be provided for lunch, at no additional cost.

Register for LMT at https://lhsmath.org/LMT/Registration/Home.

Girls’ LMT 2025 will be held ONLINE on MathDash from 11:00 AM-4:15 PM EST on Saturday, April 19th, 2025. Participation is open to middle school students who identify as female or non-binary. The competition will include an individual round and a team round with a break for lunch and mini-events. It is free to participate.

Register for GLMT at https://www.lhsmath.org/LMT/Girls_LMT.

More information is available on our website: https://lhsmath.org/LMT/Home. Email lmt.lhsmath@gmail.com with any questions.
20 replies
vrondoS
Mar 27, 2025
ehe
Today at 4:17 PM
2016 Sets
NormanWho   108
N Today at 4:12 PM by akliu
Source: 2016 USAJMO 4
Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set ${1, 2,...,N}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.
108 replies
NormanWho
Apr 20, 2016
akliu
Today at 4:12 PM
AMC 10/AIME Study Forum
PatTheKing806   113
N Today at 4:08 PM by rayford
[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!
113 replies
PatTheKing806
Mar 27, 2025
rayford
Today at 4:08 PM
Vertices of a pentagon invariant: 2011 USAMO #2
tenniskidperson3   51
N Today at 2:45 PM by akliu
An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.
51 replies
tenniskidperson3
Apr 28, 2011
akliu
Today at 2:45 PM
Orange MOP Opportunity
blueprimes   16
N Today at 2:44 PM by fruitmonster97
Hello AoPS,

A reputable source that is of a certain credibility has communicated me about details of Orange MOP, a new pathway to qualify for MOP. In particular, 3 rounds of a 10-problem proof-style examination, covering a variety of mathematical topics that requires proofs will be held from September 27, 2025 12:00 AM - November 8, 2025 11:59 PM EST. Each round will occur biweekly on a Saturday starting from September 27 as described above. The deadline for late submissions will be November 20, 2025 11:59 PM EST.

Solutions can be either handwritten or typed digitally with $\LaTeX$. If you are sending solutions digitally through physical scan, please make sure your handwriting is eligible. Inability to discern hand-written solutions may warrant point deductions.

As for rules, digital resources and computational intelligence systems are allowed. Textbooks, reference handouts, and calculators are also a freedom provided by the MAA.

The link is said to be posted on the MAA website during the summer, and invites aspiring math students of all grade levels to participate. As for scoring, solutions will be graded on a $10$-point scale, and solutions will be graded in terms of both elegance and correctness.

As for qualification for further examinations, the Orange MOP examination passes both the AIME and USAJMO/USAMO requirement thresholds, and the top 5 scorers will receive the benefits and prestige of participating at the national level in the MOP program, and possibly the USA TST and the USA IMO team.

I implore you to consider this rare oppourtunity.

Warm wishes.
16 replies
1 viewing
blueprimes
Today at 3:24 AM
fruitmonster97
Today at 2:44 PM
An identity in a right triangle.
Virgil Nicula   2
N Oct 4, 2006 by Virgil Nicula
Source: From CRUX.
Let $ABC$ be a triangle for which $a>b$ and $a>c.$ Prove that :

$\boxed{\ A=90^{\circ}\Longleftrightarrow (\sqrt{a+b}+\sqrt{a-b})\cdot (\sqrt{a+c}+\sqrt{a-c})=2p\sqrt 2\ }\ ,$

where ${2p=a+b+c\ }\ .$
2 replies
Virgil Nicula
Oct 4, 2006
Virgil Nicula
Oct 4, 2006
An identity in a right triangle.
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Source: From CRUX.
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Virgil Nicula
7054 posts
#1 • 2 Y
Y by Adventure10, Mango247
Let $ABC$ be a triangle for which $a>b$ and $a>c.$ Prove that :

$\boxed{\ A=90^{\circ}\Longleftrightarrow (\sqrt{a+b}+\sqrt{a-b})\cdot (\sqrt{a+c}+\sqrt{a-c})=2p\sqrt 2\ }\ ,$

where ${2p=a+b+c\ }\ .$
This post has been edited 3 times. Last edited by Virgil Nicula, Oct 4, 2006, 4:23 AM
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me@home
2349 posts
#2 • 1 Y
Y by Adventure10
Not sure what you mean by [aopsnowrap][math=inline]$p$[/math],[/aopsnowrap] buy here is what I got:
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Virgil Nicula
7054 posts
#3 • 1 Y
Y by Adventure10
And reciprocally, i.e. $\Longleftarrow\ ?$
Only for Me@home.This problem is mine. And my solution has one trick, but which works with the chain of equivalencies !

Proof. Denote $2p=a+b+c\ ,\ 2\Delta=b^{2}+c^{2}-a^{2}$ and $E=(\sqrt{a+b}+\sqrt{a-b})(\sqrt{a+c}+\sqrt{a-c})\ .$
Prove easily that the following interesting chain of identities (here is the trick):
$\boxed{\ 2p^{2}-(a+b)(a+c)=2(p-a)^{2}-(a-b)(a-c)=2(p-b)^{2}-(a-b)(a+c)=2(p-c)^{2}-(a+b)(a-c)=\Delta\ }\ \ (*)\ .$
The relations $(*)$ solve immediately the our proposed problem !
$1.\blacktriangleright \Delta<0\Longrightarrow \sqrt{(a+b)(a+c)}> p\sqrt 2\ ,\ \sqrt{(a-b)(a-c)}>(p-a)\sqrt 2$ a.s.o $\Longrightarrow E> 2p\sqrt 2\ .$
$2.\blacktriangleright \Delta>0\Longrightarrow \sqrt{(a+b)(a+c)}<p\sqrt 2\ ,\ \sqrt{(a-b)(a-c)}<(p-a)\sqrt 2$ a.s.o. $\Longrightarrow E<2p\sqrt 2\ .$
$3.\blacktriangleright \Delta=0\Longrightarrow \sqrt{(a+b)(a+c)}=p\sqrt 2\ ,\ \sqrt{(a-b)(a-c)}=(p-a)\sqrt 2$ a.s.o. $\Longrightarrow E=2p\sqrt 2\ .$
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