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
Consider a grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.
Proposed by Zhao Yu Ma and David Lin Kewei, Singapore
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 .
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 (solved by Math-lover1): Positive integers and satisfy ,, and . If the number of possible ordered pairs is equal to , compute the remainder when is divided by .
Problem 4 (solved by CubeAlgo15): 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 (solved by Math-lover1): Let be a quadratic polynomial with a positive leading coefficient. There exists a positive real number such that and for . Compute .
Problem 10 (solved by aaravdodhia): Find the number of ordered triples of positive integers such that and is a multiple of .
Challenge: Make as many positive integers from 2 zeros
Biglion45
NToday at 3:25 AM
by defiw
How many positive integers can you make from at most 2 zeros, any math operation and cocatination?
New Rule: The successor function can only be used at most 3 times per number
Starting from 0, 0=0
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 and Solved Problems
1a:
ehz2701 (10; 1-10)
2a:
ehz2701 (5;1-4,19)
vanstraelen (6; 5-10)
2b:
vanstraelen (4;1,2old,3-4)
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.
A bar of chocolate has every square colored with red, green or blue uniformly at random each with probability. All borders between squares of different colors are cut, leaving several connected pieces of squares all of which are the same color. Find the expected value of the number of pieces.
In scalene triangle , the circumcentre and incentre are respectively and .
Let be the altitude to line , with lying on line . Given that the radius of the
circumcircle and A-excircle are equal, prove that the points ,, and are collinear.
Let be the assertion .
Swapping , we get that for all where is some constant, that is, for all . so and for all .
Let be sufficiently large, then for all and , so:
So is a cubic polynomials, however since for all it cannot be a cubic polynomial. Hence no solutions.
Let denote the assertion
Fixing and varying we see that is surjective
Let for some , then thus is bijective
To be continued
Ι found using the substitution jasper chose above. Too lazy to writeup.
This post has been edited 1 time. Last edited by GreekIdiot, May 17, 2025, 9:49 AM
Hi.
And therefore by switching and comparing,
which implies
Take function of both sides, and substitute into , get
Cube both sides and apply some stuff, now we havefor all . This is equivalent to . Fix a and let and . Both and are positive. Then we have . When we factor, we getNotice that, when increasing , the LHS gets arbitrarily large (and therefore not constant) if , and gets arbitrarily small (and therefore not constant) if . The only possibility is and therefore . Therefore,This implies is bijective. If we take , then . Now can be anything greater than . Because is surjective, can be anything and is strictly increasing.
If, for some we have , then contradicting the fact that is strictly increasing. If, for some we have , then which also contradicts strictly increasing. Therefore for all , and it works.
This post has been edited 7 times. Last edited by maromex, May 17, 2025, 8:16 PM Reason: Done!