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 be positive integer and fix distinct points on a circle. Determine the number of ways to connect the points with arrows (oriented line segments) such that all of the following conditions hold: [list] [*]each of the points is a startpoint or endpoint of an arrow; [*]no two arrows intersect; and [*]there are no two arrows and such that ,, and appear in clockwise order around the circle (not necessarily consecutively). [/list]
Let be a triangle with incentre , and let be the circumcircle of triangle . Let be a point in the interior of segment such that . The angle bisector of intersects at points and such that and lie on the same side of , and the angle bisector of intersects at points and such that and lie on the same side of .
Let be a cyclic quadrilateral such that and . Point lies on the line through parallel to such that and lie on opposite sides of line , and . Point lies on the line through parallel to such that and lie on opposite sides of line , and .
Prove that the perpendicular bisectors of segments and intersect on the circumcircle of .
Let be an acute triangle with , and let be the circumcircle of . Points and lie on so that and meet on the external angle bisector of . Suppose that the tangents to at and intersect at a point on the same side of as , and that and intersect at and , respectively. Let be the centre of the excircle of triangle opposite the vertex .
A real number is called a "good number" if there exists a sequence of integers such that for any positive integer ,, and is even if and only if is even.
(1) Prove that if is a good number, then and is not an integer.
(2) Prove that there are infinitely many distinct good numbers. Proposed by Dong Zichao and Wu Zhuo
Let be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers , where and is one of the integers on the board, and then erasing every integer on the board such that . The game continues until the board is empty. The player who erases the last integer on the board loses.
Determine all values of for which Geoff can ensure that he wins, no matter how Ceri plays.
Source: European Girls’ Mathematical Olympiad-2014 - DAY 2 - P5
Let be a positive integer. We have boxes where each box contains a non-negative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called solvable if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.
ok lets see if i can make a good problem that actually has a good solution to it lol (i havent done it in a while)
i have distinct colors, and I want to color dots in an array of dots, where . However, I am not allowed to color in one dot a specific color, and then color in a dot that is adjacent to the previously colored spot (diagonally, horizontally, vertically). What is the smallest that enables me to color in all of the dots successfully, and how many ways can I color that array in?
an example of an unseccessful array is shown here:
A B C
A B C
A B C
the same colors are adjacent to each other vertically.
a sucessful array is shown here (i modified it so that it wont be alligned with the problem, but i think you get the idea)
A B C
D E A
B C D
as you can see, no 2 dots of the same color touch one another (diagonally, horizontally, vertically)
I saw some users making comments about how to study from Introduction to Geometry, so I thought I'd put something down for Intermediate Algebra as well. Chapters 1, 2, and 4 are extremely important review; if you think you have completely mastered the topic, these chapters are skippable but if you have not completed Introduction to Algebra, they don't take too long and are very good review. Chapter 3 is not very important: reference Precalculus for deeper complex number studies. Chapter 5, as stated, is mostly unimportant but quite interesting. Do this if you have time, (though 5.3 is the only really important section), but it's not vitally important. Chapter 6 is one of those not negligible but not very important chapters: skim through this chapter lightly, since it really only gives quite basic information. Chapters 7 and 8 are very important: Personally, Chapter 8 is the chapter more directed towards contests and Chapter 7 is a standard Algebra II chapter; the concepts are simple, but be sure to attempt all the challenge problems. Chapter 9 is interesting but not very useful. Some of the concepts are interesting, but this chapter is not that important. Chapters 10 and 11 are the most important chapters up to this point: Chapter 10 is arguably the most important chapter in the book unless you are preparing for contests. Chapter 11 is very general, a great tool for polishing up your bashing skills. Chapter 12 is the hardest chapter in the book (excluding 17-20), and was my personal favorite chapter. Inequalities are important when training for olympiads, but if you're only aiming for AMC/AIME, it's not as important as Chapters 10, 11, etc. However, if you have time and enough persistence, the chapter is very useful (and in my opinion, one of the best-written chapters in the book.) Chapters 13-16 all address a type of function: chapter 14 is the shortest chapter in the book, and the least important of the four; chapters 13 and 16 are the most important, with logarithmic identities and custom piecewise functions being a huge topic, especially in the AIME. Chapter 15 is a standard-oriented chapter, but some of the challenge problems are quite interesting. Chapter 17 is the supplement to Chapter 10: if you're preparing for the AIME, Chapter 17 is just about as important as Chapter 10; the recursion section (17.1) is especially important. Chapter 18 is the supplement to Chapter 12: the topics discussed here will likely not be used until the USAMO, so this chapter is probably the least important of the last four chapters. It is the second-hardest chapter in the entire book. Chapter 19 is an interesting chapter; it is the only one of the last four chapters to bring up new material. Functional Equations are very high-level concepts: they frequently appear in the final AIME problems and various olympiads. Study this chapter, but you're not likely to completely understand it until you master the other chapters (and the AMC/AIME.) Chapter 20 is the "monster" of all the chapters; this chapter is the most important and hardest of all the chapters when preparing for contests. Symmetry is probably most important here, though the problems given are insufficient for you to master symmetry. Attempt all problems in this chapter.
Final Note: Don't skim over sections and only do problems you think are sufficient. Do every single problem presented in the book, and all Review and Challenge Problems. 1,100 problems still might not be sufficient for you to master all the topics, and don't be intimidated by problems sourced from olympiads, USAMO, and IMO (personally, the last few challenge problems from Chapter 12 were way easier than their source: 12.77 was extremely easy, 12.80 was literally an easier version of 12.79).
Edit:
Most important chapters for AMC 10/12: Chapters 1, 2, 4, 8, 10, 13, 20 (maybe)
Most important chapters for AIME: Chapter 2, 8, 10, 12, 13, 16, 17, 20
Most important chapters for olympiads: Chapters 2, 8, 10, 11, 12, 13, 16, 17, 18, 19, 20
(moved from MSM with modifications, if this is too hard go to MSM for an easier ver.)
(pls don’t flame me if this is too hard, I thought of it in the shower).
people this came up in the shower I was considering it and got a loosely defined statement on paper and copied it up here, please if there are errors don’t be mad. yall chilllll
To “AIMEify” a number is to convert it into , given . To AIMEify a number is to convert it into , where is power free (i.e. not divisible by the power of any prime). Define a function to AIMEify any square-root radical and any rational number such that it must always output a 3-digit number from to , inclusive.
Find all such that there exists a real number such that , or prove none exist.
(I haven’t even solven this :rotfl: so feel free to cook)
Problem 1
a) Let such that . Compute the value of the expression:
b) Let such that Prove that .
Problem 2
a) Let be a prime number greater than 5. Prove that is divisible by 240.
b) Do there exist integers such that
\item[c)] Find all natural number pairs such that Problem 3
a)Let such that . Prove that
b) Let . Prove that Problem 4
Let triangle be acute with . Let the altitudes and intersect at . The line through perpendicular to intersects the extensions of and at points and , respectively. The line through perpendicular to intersects the extensions of and at points and , respectively. Let be the midpoint of .
a) Prove that triangles and are similar.
b) Prove that triangle is isosceles and that .
c) Prove that .
Problem 5
Mr. Quý chooses two distinct integers and , then writes down the following six numbers: Among these six numbers, what is the maximum number of perfect squares that can appear? Explain why.
I would assume that the intent of the problem is to express with and an no other logarithms. This is what I did.
I'm going to brush over some of the log identities and manipulations. If anybody wants me to explain, I can. Let and . Then and We'll find and in terms of and , which will then give us Rearranging, we get and Plugging into , we get Then , giving
There is probably an algebra mistake in there somewhere.
Surely there is a more direct way to find this (though my answer doesn't hint at any). Can anybody find it?