ka April Highlights and 2025 AoPS Online Class Information
jlacosta0
Apr 2, 2025
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)!
Prealgebra 1
Sunday, Apr 13 - Aug 10
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29
Introduction to Algebra A
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28
Introduction to Counting & Probability
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19
Introduction to Number Theory
Thursday, Apr 17 - Jul 3
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30
Introduction to Algebra B
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14
Introduction to Geometry
Wednesday, Apr 23 - Oct 1
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19
Intermediate: Grades 8-12
Intermediate Algebra
Monday, Apr 21 - Oct 13
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22
MATHCOUNTS/AMC 8 Basics
Wednesday, Apr 16 - Jul 2
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21
AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22
Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22
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).
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.
-----------------------------
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.
Disclaimer: this entire post is my opinion based solely on my own experiences. I happen to do a lot of teaching, so if you have thoughts or opinions of your own, please do share - I could use them. :)
Recently I've seen a bunch of improvement threads come up which I feel can be answered in more or less the same way. Rather than post a reply to each of these I decided I'd just make a separate topic and hope people read/link it.
I'm going to start by reposting something I wrote at the end of last year: [quote="v_Enhance"]In general, I think once you figure out what you're trying to improve then it doesn't make a big difference whether you do ARML vs Mandelbrot vs NIMO vs OMO or whatever.[/quote] In retrospect I should have bolded this sentence, because it was one of the main points I was trying to make: the choice of what book/problems you choose to do really doesn't matter. The only book I've ever read more than half of was Volume 2. WOOT was helpful when I started really getting serious about olympiads because it filled in various knowledge gaps (e.g. complex numbers), but beyond that my training basically consists of doing random problems from the contest section. (I don't even bother to pick topics.)
This is why I feel the extensive discussion over "which chapters of Volume 1 are most important", "which tests are about the same difficulty as X", "breakdown of AMC10 topics", et cetera, get more attention than they deserve. Really, I think that once you know the fundamentals -- and there aren't that many -- only a few things actually make a difference during practice:
[list=1]
[*]Difficulty of problems. Specifically, do hard problems when you get the chance. Doing easy problems makes you faster and less prone to errors but does nothing to help expand the range of problems you can solve. So, do hard problems, and don't worry about time until later. (I have fond memories of USAMTS for that.) Of course, this is something that should be done during the academic year, and not right before a major contest.
[*]Building intuition. See this. Particularly the third point, which I think applies in full force to any level of competition. The really important thing to see is that people don't pull solutions to hard problems out of thin air; there's a thought process behind it all. MOPpers don't solve problems just because they were born with some innate talent; they've built an intuition from doing lots and lots of hard problems.[/list]
Beyond that it just comes down to doing enough problems. Speed/careless issues are things that you should practice for sometimes, but not most of the time, and definitely not exclusively.
In closing, some little things for people who are just starting out with contests:
[list=1]
[*]Don't cram. Just relax on the couple days before the contest. I find Starbucks Coffee a good, free way to relax.
[*]Don't take contests too seriously. For me the best part about these contests has always been getting to know other really cool people, and we're all getting together this weekend![/list]
I was docked 4 points on jmo 1 and it cost me mop. I got 370 777 and 770 777 got into mop.
This google drive link contains my submission to USAJMO day 1 Day 1 Scans
My solution works except for 2 typos. I wrote bijective instead of non bijective at the end, and i wrote min intead of more specifically minimum over Z. After discussion with vsamc, and megarnie they agreed I should have gotten a 7 on this problem because i demostrated that I knew how to solve it. Is it possible to protest my score, and get into MOP.
text totally not copied over from wmc (thanks jason <3) Quick Links:
[list=disc]
[*] National:(Sprint)(Target)(Team)(Sprint + Target Submission)(Team Submission) [/*]
[*] Miscellaneous:(Leaderboard)(Sprint + Target Private Discussion Forum)(Team Discussion Forum)[/*]
[/list]
----- Eddison Chen (KS '22 '24), Aarush Goradia (CO '24), Ethan Imanuel (NJ '24), Benjamin Jiang (FL '23 '24), Rayoon Kim (PA '23 '24), Jason Lee (NC '23 '24), Puranjay Madupu (AZ '23 '24), Andy Mo (OH '23 '24), George Paret (FL '24), Arjun Raman (IN '24), Vincent Wang (TX '24), Channing Yang (TX '23 '24), and Jefferson Zhou (MN '23 '24) present:
[center]IMAGE[/center]
[center]Image credits to Simon Joeng.[/center]
2024 MATHCOUNTS Nationals alumni from all across the nation have come together to administer the first-ever ELMOCOUNTS Competition, a mock written by the 2024 Nationals alumni given to the 2025 Nationals participants. By providing the next generation of mathletes with free, high quality practice, we're here to boast how strong of an alumni community MATHCOUNTS has, as well as foster interest in the beautiful art that is problem writing!
The tests and their corresponding submissions forms will be released here, on this thread, on Monday, April 21, 2025. The deadline is May 10, 2025. Tests can be administered asynchronously at your home or school, and your answers should be submitted to the corresponding submission form. If you include your AoPS username in your submission, you will be granted access to the private discussion forum on AoPS, where you can discuss the tests even before the deadline.
[list=disc]
[*] "How do I know these tests are worth my time?"
See our credentials at the top of the post - every member of our problem-writing committee is an alumnus of the 2024 MATHCOUNTS National Competition. Additionally, many of us have written problems for previous contests such as WMC.
[/*]
[*] "Who can participate?"
Anyone may participate in any stage of the competition, regardless of age or their previous scores!
[/*]
[*] "How do I sign up?"
Signing up is neither mandatory nor binding, but it adds this thread to your feed for update, as well as shows us support :). You can do so in a reply to this thread by adding your username to the list and incrementing the count.
[/*]
[*] "What if I have multiple students?"
You can administer all of them together, then have each student submit their answers separately. For the Team round (if any, see below), have the captain submit the team's answers once. One school may have as many teams as they like.
[/*]
[*] "What if a problem is ambiguous, incorrect, etc.?"
If you are taking the test and think there is an error in one of the problems, skip the problem and keep going (as you would in an actual competition). After the timer goes off, email us at elmocounts2025@gmail.com (do not spoil it in this thread!).
[/*]
[*] "Will there be solutions?"
Official solutions will be released publicly after the corresponding submission deadline. In the meantime, you can see what others have shared in the private discussion forum. If you're still stuck on one of the problems, feel free to ask in the forum or email us at elmocounts2025@gmail.com and we'd be happy to help.
[/*]
[*] "Will there be a Countdown Round administered?"
If you all are interested in having a Countdown Round administered at the 2025 National Competition, let us know below or in the form - if enough people are interested, we may write a Countdown Round.
[/*]
[/list]
If you have any other questions, feel free to email us at elmocounts2025@gmail.com (or PM me)!
Hi everyone,
I just have a question. I live in PA and I sadly didn't make it to nationals this year. Is PA a competitive state? I'm new into mathcounts and not sure
Let be a positive integer such that , and let the set consist of distinct positive integers not exceeding 2025. Suppose that for every , with , if , then as well. Prove that
Aiming for AIME, then JMO afterwards, is Alcumus adequate, or I still need to do the problems on AoPS books?
I got AMC 23 this year, and never took amc10 before. If I master the alcumus of intermediate algebra ( making all of the bars blue). How likely I can qualify for AIME 2026?
I believe discussion is allowed after yesterday at midnight, correct?
If so, I will put tentative answers on this thread.
By the way, does anyone know the answer to Geometry Problem 5? I was wondering if I got that one right
Also, if you put answers, please put it in a hide tag
Answers for the Algebra Subject Test
1.
2.
3.
4.
5.
6.
7. (i dont remember exactly if it was 8032038 or some other number, so correct me on this)
8. (i posted this at 12:29) lol
9. ????
10.???
Tiebreaks: ???
Estimated Algebra Cutoffs
Top 50%:
3+tiebreaks, 4 guarantees?
Top 20%
5+tiebreaks, 6 guarantees?
Top 10:
8 + tiebreaks, 9 guarantees?
for these lmk if ur opinion differs this is only my opinion
2024 PUMaC Team Round Question 14 reads as follows:
What is the largest value of for which I can find nonnegative integers such that for all indices , divides ?
(Note: This should say "... nonnegative integers ...")
I interpreted this correction to mean the following:
What is the largest value of for which I there exists nonnegative integers such that for all indices , divides ?
I checked this with Python using the following code:
[code]def digit_sum_base(n, base):
total = 0
while n > 0:
total += n % base
n //= base
return total
target_sum = 19
base = 17
limit = 2024
qualified_numbers = [n for n in range(limit) if digit_sum_base(n, base) == target_sum]
qualified_numbers.append(2023)
from math import comb
all_divisible = True
for i in range(len(qualified_numbers)):
for j in range(i):
a, b = qualified_numbers, qualified_numbers[j]
if comb(a, b) % 17 != 0:
all_divisible = False
break
if not all_divisible:
break
1. Let and be positive integers. Prove that there exists a positive integer such that for every odd integer , the digits in the base- representation of are all greater than .
Rubric for Problem 1
Solution: We prove this constructively.
Lemma 1: has at most digits in its base- representation. Proof: For the base representation of to have more than digits, we must have . But this implies , which is clearly false for positive.
1 point for proving this lemma.
Fix . Let the (unique) base- representation of be . Define and to be the remainder when is divided by . Notice that and is odd for . The solution then hinges on the following construction: or an equivalent formulation. To prove this construction works, we need to show that
[list]
[*] is always an integer between and inclusive.
[*]This construction is valid.
[*]All can become arbitrarily large.
[/list]
1 point for having a construction that uses floors and remainders.
Notice that by definition, is the remainder when is divided by . Thus is always an integer. Additionally, since , each coefficient is between and .
1 point for showing that is always an integer between and inclusive in the construction chosen.
Notice that this sum equals since it telescopes.
2 points for showing the construction chosen evaluates to .
Finally, since , we have Thus, each digit is lower bounded by , which can become arbitrarily large as becomes arbitrarily large.
2 points for showing that each digit is lower bounded by a value that can become arbitrarily large.
Remark: Deduct 1 point if a value for is given but some fails.
2. Let and be positive integers with . Let be a polynomial of degree with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers such that the polynomial divides , the product is zero. Prove that has a nonreal root.
Rubric for Problem 2
Solution: Suppose had no nonreal roots. We can assume has degree , as we can always find a polynomial such that has no nonreal roots. We can also assume is monic by scaling. Also, the case is trivial as the constant term must be nonzero, so fix . Let the roots of be .
1 point for reducing to .
Consider the degree polyonmials which has the roots for all integers . Let be the coefficient of the term of . Then, we have for all . Since , by Pigeonhole Principle, there exists a value of such that there exists two values of for which .
2 points for using Pigeonhole Principle in some manner on roots and degree polynomials.
Suppose WLOG that the two values of are and . Consider the polynomial with roots . Then we know that the coefficient of the term is in both and .
If is the coefficient of the term in , then expanding gives and . Since , solving this gives . Since , has two consecutive nonleading coefficients equal to .
2 points for concluding some polynomial dividing has two consecutive coefficients equal to zero. Some approaches will lead to a polynomial with three consecutive nonzero coefficients in geometric progression (possibly with ratio ). If this is the case, reward only 1 point.
Lemma 1: A polynomial with two consecutive nonleading coefficients equal to cannot have all distinct real roots. Proof: By Rolle's Theorem, the derivative has distinct real roots as well, along with two consecutive nonleading coefficients equal to zero, but one degree lower, by the Power Rule. By doing this repeatedly, we eventually end up with a polynomial where the two consecutive coefficients have degree and respectively. But this polynomial has a double root at , contradiction.
2 points for proving this lemma.
3. Alice the architect and Bob the builder play a game. First, Alice chooses two points and in the plane and a subset 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 of cities, they are connected with a road along the line segment if and only if the following condition holds:
[center]For every city distinct from and , there exists such[/center]
[center]that is directly similar to either or .[/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: is directly similar to if there exists a sequence of rotations, translations, and dilations sending to , to , and to .
Rubric for Problem 3
Solution: Alice wins by taking to be the set of points strictly outside the circle with diameter .
2 points for claiming Alice wins with the correct subset . No points should be rewarded for just claiming Alice wins.
Lemma 1: No two roads can cross. Proof: The region Alice has chosen means that is always acute. Suppose two roads and cross. Then, is a convex quadrilateral. Since is a quadrilateral, one of its angle is not acute, WLOG . Then cannot be a road, as is not acute but there does not exist an such that is not acute, contradiction.
1 point for attempting to use angles in a connectivity argument. 1 additional point for completing the argument.
Now we show by induction that any two points are connected by some path.
1 point for mentioning induction on distance between points for connectivity. No points should be rewarded for just the mention of induction.
Suppose any two points with a distance are connected by some path. This is trivially true for , as no points are a distance of apart. We show that any two points with a distance of are also connected.
Suppose and are a distance of apart. If there are no points in the circle of diameter , then there is a road between and . Otherwise, there is a city in the circle. Observe that since is obtuse, we have . Since , we must have , so then is connected to , which is connected to , done.
2 points for completing the induction argument. Deduct 1 point if the base case or is not mentioned. Do not reward points if the induction argument is incomplete or incorrect.
4. Let be the orthocenter of acute triangle , let be the foot of the altitude from to , and let be the reflection of across . Suppose that the circumcircle of triangle intersects line at two distinct points and . Prove that is the midpoint of .
Rubric for Problem 4
IMAGE
Deduct 1 point if a diagram is missing.
Solution 1: Let be the center of . Let be the intersection of and the line through parallel to . Since , lies on . Additionally, is the -antipode of .
2 points for constructing and identifying it lies on . 1 additional point for identifying it as the -antipode of .
Then, we have . Let be the foot of the altitude from to . Additionally, since and , is a midline and thus .
Then is a midline of , so and thus .
2 points for identifying . Deduct 1 point if insufficient explanation is given for the equal lengths.
Since and (by definition), by either HL congruence or the Pythagorean theorem, we must have .
2 points for drawing this conclusion. Deduct 1 point if insufficient explanation is given for going from to . Do not deduct more than 1 point total for insufficient explanations throughout the entire solution.
Other solutions (including bashes): Since there are many approaches to this problem, for incomplete solutions, reward points as follows.
[list]
[*]1 point for identifying and proving/citing lies on .
[*]1 point for using the perpendicular bisector of both and to attempt to identify .
[/list]
If the solution attempts to construct from the perpendicular bisectors, then prove that , reward points as follows.
[list]
[*]1 point for constructing the midpoint of .
[*]1 point for constructing the nine point center of and proving etiher or , or something of similar nature.
[*]1 point for showing .
[/list]
If the solution attempts to construct such that , then prove that lies on the perpendicular bisectors, reward points as follows.
[list]
[*]1 point for constructing the midpoint of .
[*]1 point for showing that lies on the perpendicular bisector of (or showing that .
[*]1 point for showing that lies on the perpendicular bisector of (or showing that .
[/list]
The rest of the solution finishes as shown in Solution 1.
5. Determine, with proof, all positive integers such that is an integer for every positive integer .
Rubric for Problem 5
Solution: The answer is all even integers . For , we have is divisible by , which is not true for odd.
1 point for stating the correct answer and showing that odd fail.
Let be a prime power dividing . Notice that by definition we have . Since , we have if and otherwise, so for each term, either both the numerator are divisible by , or neither are.
For each term such that , we have , so . This is valid since has an inverse modulo . For each term such that , we can divide out a from both the numerator and the denominator. Notice that what's left is simply . Thus, we conclude that 2 points for expressing modulo or in this form.
We will now use induction on . For , we clearly have
Now consider where , and suppose that satisfies the induction hypothesis for the prime . Clearly . Then we have By the induction hypothesis, divides the inner binomial sum, so since we are multiplying it by , must divide .
For all integers , all even work for every . Thus, all even works for all integers .
4 points for a complete induction. Deduct 1 point if the synthesis of primes is not mentioned. Deduct 1 point if the base case is missing. Deduct 2 points if both are missing. If the induction is only done for prime powers of instead of all multiples of , reward only 1 point.
Remark: If the expression of is incorrect, reward up to points total.
6. Let and be positive integers with . There are cupcakes of different flavors arranged around a circle and 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 , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least . Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .
Rubric for Problem 6
Solution: Call each person's partitions their bubbles. We can make the following generalization: for each person , reduce their score on each cupcake by some nonnegative real number so that each bubble's total score is exactly . This would imply the original problem.
Consider one of people, Evan. Draw a bipartite graph between the set of people except Evan and the set of Evan's bubbles , where a person is connected to a bubble if the total score of the cupcakes in that bubble with respect to is at least .
1 point for setting up this bipartite graph.
Now we will apply Hall's Marriage Lemma. Hall's Marriage Lemma states that there are two cases:
Case 1: There does not exist a subset of such that . Here, denotes the set of neighbors of . Then, it is possible to match each person in with a bubble in such that they are connected. Case 2: This is not the case.
1 additional point for mentioning Hall's Marriage Lemma. Do not reward this point if the proper bipartite graph is not set up.
Lemma 1: If a person (not Evan) is not connected with a bubble , then if a different person takes the bubble, can join together two of their bubbles and still satisfy the hypothesis. Proof: Since is not connected with , their score for that bubble is less than . Thus cannot entirely contain any of 's bubbles, so overlaps with at most two of 's bubbles. Finally, since the combined score of these two bubbles is , removing the cupcakes of takes away a score of less than , so can combine these two bubbles into one large bubble with a score of at least . If overlaps with only one bubble, arbitrarily join that bubble with a neighboring one. This is a reduction from people.
2 points for observing this reduction step.
First, notice that . We now apply the following reduction algorithm:
If case 2 applies, there is a bad subset for which the people in cannot all be satisfied. Remove the set and from the graph, noting that no person in is connected to any set not in . Reapply Hall's Marriage Theorem until either case 1 applies or the set becomes empty. Notice that throughout this process, no bubble can be connected to a removed person by definition.
When case 1 applies, we can match each person with a bubble such that all of them are satisfied. By Lemma 1, this reduction step is valid, as any person removed is not connected to any bubble not removed. Otherwise, the set of people eventually becomes empty, as more people are removed than bubbles at each removal. Then, we can match Evan with an empty bubble, which is again valid by Lemma 1. In either case, the problem is reduced to fewer people.
If case 1 immediately applies, then we are done. Otherwise, the problem is eventually reduced to one person. When , the problem is trivial, as they can just take all cupcakes.
3 points for completing the reduction argument using Hall's Marriage Lemma. Deduct 1 point if the argument is complete but the case is not mentioned. 1 point may be rewarded if the argument is incomplete or incorrect but a reduction using the two cases of Hall's Marriage Lemma is seriously attempted.
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Note that in order for ,. Therefore we get
Multiply both sides by (we don't care about the restriction at ) to get . By the trivial inequality this will always be satisfied for values of .
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Note that in order for ,. Therefore we get
Multiply both sides by (we don't care about the restriction at ) to get . By the trivial inequality this will always be satisfied for values of .