ka May Highlights and 2025 AoPS Online Class Information
jlacosta0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.
Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.
Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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)!
Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.
Introduction to Algebra A
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
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
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
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14
Introduction to Geometry
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
Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Intermediate: Grades 8-12
Intermediate Algebra
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
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
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
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.
p1. Square has side length . Let the midpoint of be . What is the area of the overlapping region between the circle centered at with radius and the circle centered at with radius ? (You may express your answer using inverse trigonometry functions of noncommon values.)
p2. Find the number of times occurs when for the function .
p3. Stanford is building a new dorm for students, and they are looking to offer room configurations: Configuration : a one-room double, which is a square with side length of , Configuration : a two-room double, which is two connected rooms, each of them squares with a side length of .
To make things fair for everyone, Stanford wants a one-room double (rooms of configuration ) to be exactly m larger than the total area of a two-room double. Find the number of possible pairs of side lengths , where ,, such that .
p4. The island nation of Ur is comprised of islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands and if each island chooses the other). Then, all islands connected by bridges together form an island-state. What is the expected number of island-states Ur is divided into?
p5. Let and be the roots of the polynomial . Compute .
p6. Carol writes a program that finds all paths on an 10 by 2 grid from cell (1, 1) to cell (10, 2) subject to the conditions that a path does not visit any cell more than once and at each step the path can go up, down, left, or right from the current cell, excluding moves that would make the path leave the grid. What is the total length of all such paths? (The length of a path is the number of cells it passes through, including the starting and ending cells.)
p7. Consider the sequence of integers an defined by , for prime and for . Find the smallest such that is a perfect power of .
p8. Let be a triangle whose -excircle, -excircle, and -excircle have radii ,, and , respectively. If and the perimeter of is , what is the area of ?
p9. Consider the set of functions satisfying:
(a)
(b) ,
(c) ,
(d) .
If can be written as where are distinct primes, compute .
p10. You are given that and that the first (leftmost) two digits of are 10. Compute the number of integers with such that starts with either the digit or (in base ).
p11. Let be the circumcenter of . Let be the midpoint of , and let and be the feet of the altitudes from and , respectively, onto the opposite sides. intersects at . The line passing through and perpendicular to intersects the circumcircle of at (on the major arc ) and , and intersects at . Point lies on the line such that is perpendicular to . Given that and , compute .
p12. Let be the isosceles triangle with side lengths . Arpit and Katherine simultaneously choose points and within this triangle, and compute , the squared distance between the two points. Suppose that Arpit chooses a random point within . Katherine plays the (possibly randomized) strategy which given Arpit’s strategy minimizes the expected value of . Compute this value.
p13. For a regular polygon with sides, let denote the regular polygon with sides such that the vertices of are the midpoints of every other side of . Let denote the polygon that results after applying f a total of k times. The area of where is a pentagon of side length , can be expressed as for some positive integers where is not divisible by the square of any prime and does not share any positive divisors with and . Find .
p14. Consider the function . This function can be expressed in the form for sequences of integers ,. Determine .
p15. In , let be the centroid and let the circumcenters of ,, and be , and , respectively. The line passing through and the midpoint of intersects at . If the radius of circle is , the radius of circle is , and , what is the length of ?
PS. You should use hide for answers. Collected here.
Source: Serbian selection contest for the IMO 2025
Let be an acute triangle. Let be the reflection of point over the line . Let and be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment . Let and be the points where the reflection of line with respect to line intersects the circumcircle of triangle , where point lies on the arc not containing . If is a point on the line such that , prove that .
Let be a triangle. Circle passes through , meets segments and again at points and respectively, and intersects segment at and such that lies between and . The tangent to circle at and the tangent to circle at meet at point . Suppose that points and are distinct. Prove that line is parallel to .
In triangle with circumcircle and incenter , point bisects arc and line meets at . The excircle opposite to touches at point . Point on the circumcircle of is such that . Prove that the lines and meet on .
Let be an inscribed circle of and touching ,, at ,, respectively. Let and be diameters of . Let and be the pole of and with respect to , respectively. cuts again at . cuts again at . The tangent at of cuts at . The tangent at of cuts at . Let and be midpoint of and , respectively.
Show that : , and perpendicular bisector of are concurrent.
A hunter and an invisible rabbit are playing again...
Phorphyrion1
N2 hours ago
by JARP091
Source: 2021 Discord CCCC P4
A hunter and an invisible rabbit play a game in a grid. The rabbit's starting square is (unknown to the hunter), and after rounds, the rabbit is at square . In the -th round of the game, two things occur in order:
(i) The rabbit moves invisibly to a square which shares a point with (There are up to eight of these).
(ii) A tracking device searches squares of the hunter's choosing. If the rabbit is in one of these squares, the rabbit is captured and the game ends.
For what can the rabbit avoid capture indefinitely?
Let be an acute triangle with orthocenter . Let be the point such that the quadrilateral is a parallelogram. Let be the point on the line such that bisects . Suppose that the line intersects the circumcircle of the triangle at and . Prove that .
Let , where is rational for . A vector is called a rational point in -dimensional space. Denote the set of all such vectors as . For and in , define the distance between points and as . We say that point can move to point if and only if there is a unit distance between two points in .
Prove:
(1) If , there exists a point that cannot be reached from the origin via a finite number of moves.
(2) If , any point in can be reached from any other point via moves.
This is very elegant by AM-GM inequality. If we make the substitution that and homogeneously the others, then we shall get: . Now, let's apply the AM-GM inequality because and the others are always nonnegative. Now if we apply this to , then we shall get that . Doing this for the second term, we get that . Similarly, . Now multiplying the geometric means gets us exactly . But remember, we had our greater than or equal symbol so we have to remember that this corresponds to saying . Therefore, equality must hold and by the properties of AM-GM, we must have that and and . Thus, . Thus, our ordered triple is .