ka July Highlights and 2025 AoPS Online Class Information
jwelsh0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!
[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]
MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.
Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.
Prealgebra 2
Friday, Jul 25 - Nov 21
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18
Introduction to Algebra A
Tuesday, Jul 15 - Oct 28
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3
Introduction to Counting & Probability
Wednesday, Jul 2 - Sep 17
Sunday, Jul 27 - Oct 19
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8
Introduction to Number Theory
Tuesday, Jul 15 - Sep 30
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2
Introduction to Algebra B
Friday, Jul 18 - Nov 14
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30
Introduction to Geometry
Monday, Jul 14 - Jan 19
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29
Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)
Intermediate: Grades 8-12
Intermediate Algebra
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
Sunday, Sep 28 - Mar 29
Wednesday, Oct 8 - Mar 8
Sunday, Nov 16 - May 17
Thursday, Dec 11 - Jun 4
Precalculus
Wednesday, Aug 6 - Jan 21
Tuesday, Sep 9 - Feb 24
Sunday, Sep 21 - Mar 8
Monday, Oct 20 - Apr 6
Sunday, Dec 14 - May 31
Advanced: Grades 9-12
Calculus
Sunday, Sep 7 - Mar 15
Wednesday, Sep 24 - Apr 1
Friday, Nov 14 - May 22
Contest Preparation: Grades 6-12
MATHCOUNTS/AMC 8 Basics
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Oct 6 - Jan 12
Thursday, Oct 16 - Jan 22
Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)
MATHCOUNTS/AMC 8 Advanced
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Tuesday, Aug 26 - Nov 11
Thursday, Sep 4 - Nov 20
Friday, Sep 12 - Dec 12
Monday, Sep 15 - Dec 8
Sunday, Oct 5 - Jan 11
Tues, Thurs & Sun, Dec 2 - Jan 11 (meets three times a week!)
Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)
AMC 10 Problem Series
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 10 - Nov 2
Thursday, Aug 14 - Oct 30
Tuesday, Aug 19 - Nov 4
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)
AMC 10 Final Fives
Friday, Aug 15 - Sep 12
Sunday, Sep 7 - Sep 28
Tuesday, Sep 9 - Sep 30
Monday, Sep 22 - Oct 13
Sunday, Sep 28 - Oct 19 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, Oct 8 - Oct 29
Thursday, Oct 9 - Oct 30
AMC 12 Problem Series
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)
AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:
To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.
More specifically:
For new threads:
a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.
Examples: Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿) Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"
b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.
Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".
c) Good problem statement:
Some recent really bad post was:
[quote][/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.
For answers to already existing threads:
d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve , do not answer with " is a solution" only. Either you post any kind of proof or at least something unexpected (like " is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.
e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.
To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!
Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).
The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
Let and be circles with centres and , respectively, such that the radius of is less than the radius of . Suppose and intersect at two distinct points and . Line intersects at and at , so that lie on in that order. Let be the circumcentre of triangle . Line meets again at and meets again at . Let be the orthocentre of triangle .
Prove that the line through parallel to is tangent to the circumcircle of triangle .
Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number which is known to both players. On the th turn of the game (starting with ) the following happens:
[list]
[*] If is odd, Alice chooses a nonnegative real number such that [*]If is even, Bazza chooses a nonnegative real number such that [/list]
If a player cannot choose a suitable , the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.
Determine all values of for which Alice has a winning strategy and all those for which Bazza has a winning strategy.
Proposed by Massimiliano Foschi and Leonardo Franchi, Italy
What i am trying to do is finding a general solution to this earlier problem (https://artofproblemsolving.com/community/u1056796h3610029p35343126) . Here i am trying to generalise such a solution for n points . Simply stated find the min value of . I would love to hear your insights on this problem .
My insights :- I am not able to move ahead but probably we will find a constraint here through some inequality which will help us reach the solution .
Let be a positive integer. There are knights sitting at a round table. They consist of pairs of partners, each pair of which wishes to shake hands. A pair can shake hands only when next to each other. Every minute, one pair of adjacent knights swaps places. Find the minimum number of exchanges of adjacent knights such that, regardless of the initial arrangement, every knight can meet her partner and shake hands at some time.
What do you think about difficulty of IMO 2025 problems?
P1: 10M - typical P1, strightforward technique but with a case that some students might miss
P2: don't know
P3: 35M - on the easier side of P3
P4: 15-20M - quite difficult for P4, can even be a middle problem confortably, much harder than IMO 2005 P4
P5: 25-30M - a little bit on the harder side of P5, comparable to IMO 2016 P5
P6: 45M - on the harder side of P6, considerably harder than IMO 2022 P6
Source: 2024 March 谜之竞赛-3, Generalization of aops.com/community/c6h3107633p28107117
Let rational number where , are coprime positive integers. Show that for any integer , there exists positive integers ,,, satisfying Created by Xianbang Wang, High School Affiliated to Renmin University of China
Title says most of it. I've been meaning to post a problem set on HSM since at least a few months ago, but since I proposed the most recent problems I made to the 2025 SSMO, I had to wait for that happen. (Hence, most of these problems will probably be familiar if you participated in that contest, though numbers and wording may be changed.) The problems are very roughly arranged by difficulty. Enjoy!
Problem 1: An sequence of positive integers has the property for every positive integer , its term is greater than the mean of the first terms, and the sum of its first terms is a multiple of . Let be the number of such sequences satisfying . Compute the remainder when is divided by .
Problem 2 (solved by fruitmonster97): Rhombus has side length . Point lies on segment such that line is perpendicular to line . Given that , the area of can be expressed as , where and are relatively prime positive integers. Compute .
Problem 3: Positive integers and satisfy ,, and . If the number of possible ordered pairs is equal to , compute the remainder when is divided by .
Problem 4: Let be a triangle. Point lies on side , point lies on side , and point lies on side such that ,, and . Let be the foot of the altitude from to . Given that ,, and , the value of can be expressed as , where and are relatively prime positive integers. Compute .
Problem 5 (solved by maromex): Anna has a three-term arithmetic sequence of integers. She divides each term of her sequence by a positive integer and tells Bob that the three resulting remainders are ,, and , in some order. For how many values of is it possible for Bob to uniquely determine ?
Problem 6: There is a unique ordered triple of positive reals satisfying the system of equations The value of can be expressed as , where and are positive integers such that is square-free. Compute .
Problem 7: Let be the set of all monotonically increasing six-term sequences whose terms are all integers between and inclusive. We say a sequence in is symmetric if for every integer , the number of terms of that are at least is . The probability that a randomly chosen element of is symmetric is , where and are relatively prime positive integers. Compute .
Problem 8: For a positive integer , let denote the value of the binary number obtained by reading the binary representation of from right to left. Find the smallest positive integer such that the equation has at least ten positive integer solutions .
Problem 9: Let be a quadratic polynomial with a positive leading coefficient. There exists a positive real number such that and for . Compute .
Problem 10: Find the number of ordered triples of positive integers such that and is a multiple of .
There is a lack of trigonometric bash practice, and I want to see techniques to do these problems. So here are 10 kinds of problems that are usually out in the wild. How do you tackle these problems? (I had ChatGPT write a code for this.). Please give me some general techniques to solve these kinds of problems, especially set 2b. I’ll add more later.
Leaderboard
1a:
ehz2701 (10; 1-10)
2a:
ehz2701 (4;1-4)
vanstraelen (6; 5-10)
2b:
vanstraelen (1;1)
problem set 1a (1-10)
Problem Set 1a. Show that
I admit 1a-6 and 1a-10 is a bit easy analytically. However, the point of the exercise is improving the ability of trigonometric identities.
Concurrency of Lines Involving Altitudes and Circumcenters in a Triangle
JackMinhHieu2
N2 hours ago
by JackMinhHieu
Hi everyone,
I recently came across an interesting geometry problem that I'd like to share. It involves a triangle inscribed in a circle, altitudes, points on arcs, and a surprising concurrency involving circumcenters. Here's the problem: Problem:
Let ABC be an acute triangle inscribed in a circle (O). Let the altitudes AD, BE, CF intersect at the orthocenter H.
Points M and N lie on the minor arcs AB and AC of circle (O), respectively, such that MN // BC.
Let I be the circumcenter of triangle NEC, and J be the circumcenter of triangle MFB.
Prove that the lines OD, BI, and CJ are concurrent.
I find the configuration quite elegant, and I'm looking for different ways to approach the problem — whether it's synthetic, coordinate, vector-based, or inversion.
Any ideas, hints, or full solutions are appreciated. Thank you!
Do your schools let you take Calc in middle school?!
well aops and rms allow it…
the highest u can take in 8th grade at my school is algebra 2 in 8th grade
for my school it's geometry... our school has some really bad fluency percents lol
same
whats actually kinda sad is how so many parents are concerned with the fact that some kids are 2-3 grade levels ahead instead of focusing on their child. It's more of a crab mentality rather than being focused about their child's future. in my middle school they were about to remove geometry because plenty of parents whose 8th graders were taking algebra were concerned.
whats actually kinda sad is how so many parents are concerned with the fact that some kids are 2-3 grade levels ahead instead of focusing on their child. It's more of a crab mentality rather than being focused about their child's future. in my middle school they were about to remove geometry because plenty of parents whose 8th graders were taking algebra were concerned.
huh, there's no need to drag down kids who are excelling
standards based grading is a grading system that uses standards and no letter grades.
no percents are implemented so its considered low stress.
imo tho
IT IS AN ABOMINATION.
like it gives me no motivation and the grades are terrible and easy
i understand that some parents want this but this just isnt the motivation and stress i want
china japan or korea has way better education system ()
Korea just has traditions of children going to tutoring (학원) and afterschool the kids just go to these tutoring for a rlly long time as almost a second school
This post has been edited 1 time. Last edited by K1mchi_, May 21, 2025, 6:54 PM
standards based grading is a grading system that uses standards and no letter grades.
no percents are implemented so its considered low stress.
imo tho
IT IS AN ABOMINATION.
like it gives me no motivation and the grades are terrible and easy
i understand that some parents want this but this just isnt the motivation and stress i want
My school system: there are letter grades, but there are decimals that correspond to letter grades with lower cutoffs (C = 1.6+, B = 2.7+, A = 3.4+)
This is an abomination as well...such low standards for grades.
In our district they emphasize A LOT how 'most students don't skip' and 'it's okay and recommended to not skip' etc
That doesn't help anyone...
whats actually kinda sad is how so many parents are concerned with the fact that some kids are 2-3 grade levels ahead instead of focusing on their child. It's more of a crab mentality rather than being focused about their child's future. in my middle school they were about to remove geometry because plenty of parents whose 8th graders were taking algebra were concerned.
Bruhhh that's like saying "Oh my child is doing bad lets make all the other kids that are doing good also be bad"
whats actually kinda sad is how so many parents are concerned with the fact that some kids are 2-3 grade levels ahead instead of focusing on their child. It's more of a crab mentality rather than being focused about their child's future. in my middle school they were about to remove geometry because plenty of parents whose 8th graders were taking algebra were concerned.
Bruhhh that's like saying "Oh my child is doing bad lets make all the other kids that are doing good also be bad"
tbf tho middle school isn’t important
still, if i can, i want to be 3 tracks ahead in prior to high school yk