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
Geometry — Orthocenter, Circle Intersections, and Parallel Lines
justalonelyguy1
Nan hour ago
by Royal_mhyasd
Let be an acute triangle with , inscribed in a circle . Let be the altitudes of triangle , and let be their intersection (the orthocenter). Let be the second point of intersection of line with circle (). Let be the second point of intersection of line with circle ().
Let be the intersection point of lines and , and let be the intersection point of lines and .
**Problem 11.** Let triangle be circumscribed about circle . The circle touches at point . A circle passing through , and tangent to at . Line intersects the circle again at point . Prove that the quadrilateral is cyclic.
For positive integer , define as the smallest positive integer satisfying the following property: for any integer coprime with , we have .
Given an integer , and integers all coprime with , prove that there exists a non-empty subset of such that Proposed by Zhenqian Peng from High School Affiliated to Renmin University of China
Let be a prime and consider a partition of the set into three disjoint subsets . Prove that there exists and all in different subsets such that is divisible by
Cheers to this being my 420th post :D! This is a collection of all of my favorite nice and cool problems I've solved on my journey so far. Enjoy! :)
Algebra: 2017 CMIMC A7: Let ,, and be complex numbers satisfying the system of equationsFind .
2019 All-Russian Olympiad Grade 10 P1: Each point in the plane is assigned a real number It is known that whenever is the centroid of Prove that for all points
1997 USAMO/5: Prove that, for all positive real numbers ,,, the inequality holds.
2016 MP4G P12: Let ,,,,, and be real numbers such that for every real number , we have Compute .
Combinatorics: Chartrand-Zhang 2.36:
Find the smallest positive integer for which there exists a simple graph on vertexes, for which exactly vertices have degree , exactly vertices have degree , and exactly vertices have degree .
NIMO 4.3: In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square away as long as it isn't occupied. One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability . Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors. The value of that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as for relatively prime positive integers . Compute .
2023 CMIMC C7: Max has a light bulb and a defective switch. The light bulb is initially off, and on the th time the switch is flipped, the light bulb has a chance of changing its state (i.e. on off or off on). If Max flips the switch 100 times, find the probability the light is on at the end.
NIMO 5.6: Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -. Tom presses a sequence of random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of . Find the expected value of . (Note: Negative numbers are permitted, so 13-22 gives . Any excess operators are parsed as signs, so -2-+3 gives and -+-31 gives . Trailing operators are discarded, so 2++-+ gives . A string consisting only of operators, such as -++-+, gives .)
2015 All-Russian Olympiad Grade 11 P5: An immortal flea jumps on whole points of the number line, beginning with . The length of the first jump is , the second , the third , and so on. The length of jump is equal to . The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flea will have been on every natural point, perhaps having visited some of the points more than once?
2008 IMO P1: Let be the orthocenter of an acute-angled triangle . The circle centered at the midpoint of and passing through intersects the sideline at points and . Similarly, define the points ,, and . Prove that the six points ,,,, and are concyclic.
1993 USAMO P2: Let be a convex quadrilateral such that diagonals and intersect at right angles, and let be their intersection. Prove that the reflections of across are concyclic.
Unknown: Consider two circles and which are internally tangent at A line intersects and at four distinct points in that order. Prove that
2009 IMO P4: Let be a triangle with . The angle bisectors of and meet the sides and at and , respectively. Let be the incentre of triangle . Suppose that . Find all possible values of .
Number Theory: 2011 USAJMO P1: Find, with proof, all positive integers for which is a perfect square.
Dudu has 729 drawers numbered from 1 to 729. Exactly one of them contains a secret document.
Dudu can open drawers to try to find it, but with the following restrictions:
Dudu can open up to 6 drawers in total.
After opening an empty drawer, Dudu receives a hint; however, this hint can be false in up to 1 of his 6 attempts (that is, when opening an empty drawer, the safe may lie and say “the document is in a drawer with a higher number” when it is actually in a lower-numbered drawer, or vice versa).
If Dudu opens the drawer containing the document, he finds it and the search ends.
(a) Show that, despite the possibility of 1 lie, it is still possible to guarantee locating the document within 6 openings.
(b) Explain the strategy that allows overcoming this lie and finding the document safely.
Let's show that has infinitely many solutions. Just rewrite then let be an even integer, , and let (we will consider later). Then, our equation becomes , which is a Pell-type equation. We can say that has infinitely many solutions with and . To have infinitely many integers, we need to have infinitely many . And that's true, since and .