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k a My Retirement & New Leadership at AoPS
rrusczyk   1571
N Yesterday at 11:40 PM by SmartGroot
I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE
1571 replies
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rrusczyk
Mar 24, 2025
SmartGroot
Yesterday at 11:40 PM
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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
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0 replies
jlacosta
Mar 2, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
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]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/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 $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ 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.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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Ahlfors 2.1.2.1
centslordm   2
N 15 minutes ago by alexheinis
If $g(w)$ and $f(z)$ are analytic functions, show that $g(f(z))$ is also analytic.
2 replies
centslordm
Jan 11, 2025
alexheinis
15 minutes ago
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Eigenvalues of A vs. f(A)
Mathloops   3
N 23 minutes ago by alexheinis
Let \( A \) be an \( n \times n \) square matrix with eigenvalues \(\lambda_1, \lambda_2, \dots, \lambda_k\) (each \(\lambda_i\) having algebraic multiplicity \( m_i \), so that \( m_1 + m_2 + \cdots + m_k = n \)). Let \( f(x) \) be a polynomial. It is known that if \(\lambda\) is an eigenvalue of \( A \) then \( f(\lambda) \) is an eigenvalue of \( f(A) \).
The question is: Are all the eigenvalues of \( f(A) \) of the form \( f(\lambda_i) \) (counting multiplicities)?

Click to reveal hidden text

Furthermore, is there any relation in the nuclear space corresponding to each of those eigenvalues? (equation $Av = \lambda v$ vs. equation $f(A)v = f(\lambda)v$)
3 replies
Mathloops
Yesterday at 3:58 PM
alexheinis
23 minutes ago
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2025 Caucasus MO Juniors P2
BR1F1SZ   1
N an hour ago by GreekIdiot
Source: Caucasus MO
There are $30$ children standing in a circle. For each girl, it turns out that among the five people following her clockwise, there are more boys than girls. Find the greatest number of girls that can stand in a circle.
1 reply
BR1F1SZ
Yesterday at 12:55 AM
GreekIdiot
an hour ago
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Unique solution
USJL   0
an hour ago
Source: 2025 Taiwan TST Round 2 Independent Study 1-N
Find all $g:\mathbb{R}\to\mathbb{R}$ so that there exists a unique $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)$ and
\[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\]for all $x,y\in\mathbb{R}$.

Proposed by usjl
0 replies
USJL
an hour ago
0 replies
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Show that XD and AM meet on Gamma
MathStudent2002   90
N an hour ago by ErTeeEs06
Source: IMO Shortlist 2016, Geometry 2
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.

Proposed by Evan Chen, Taiwan
90 replies
+1 w
MathStudent2002
Jul 19, 2017
ErTeeEs06
an hour ago
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A number theory about divisors which no one fully solved at the contest
nAalniaOMliO   7
N an hour ago by CHESSR1DER
Source: Belarusian national olympiad 2024
Let's call a pair of positive integers $(k,n)$ interesting if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$
Find the number of interesting pairs $(k,n)$ with $k \leq 100$
M. Karpuk
7 replies
nAalniaOMliO
Jul 24, 2024
CHESSR1DER
an hour ago
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Modified Fermat's last theorem
Euler8038   0
2 hours ago
Source: Own
Prove that, for any n, there is an infinite number of sequences composed by n pairwise coprime positive integers such that the sum of the n-th powers of the term in the sequence gives you an n-th power.

To be clear, if n=2 the conjecture is just about Pythagorean triples.

If n=3, you have to show that there exist an infinite number of triplets such that a³+b³+c³ is a cube, with a, b, c pairwise coprime positive integers.
0 replies
Euler8038
2 hours ago
0 replies
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Putnam 1951 A1
centslordm   3
N 2 hours ago by KAME06
Show that the determinant: \[ \begin{vmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0 & f \\ -c & -e & -f & 0 \end{vmatrix} \]is non-negative, if its elements $a, b, c,$ etc., are real.
3 replies
centslordm
May 25, 2022
KAME06
2 hours ago
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Infinite cube triplets
Euler8038   0
2 hours ago
Let a, b, x be positive coprime integers. Prove that there exist an infinite number of triplets (a, b, x) such that x³=3ab(a+b), or disprove the conjecture.
0 replies
Euler8038
2 hours ago
0 replies
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find the value of an angles
AlanLG   4
N 2 hours ago by sunken rock
Source: 1st National Women´s Contest of Mexican Mathematics Olympiad 2022, problem 2 teams
Consider $\triangle ABC$ an isosceles triangle such that $AB = BC$. Let $P$ be a point satisfying

$$\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP$$
Find all possible values of $\angle BCP$.
4 replies
AlanLG
Jul 23, 2023
sunken rock
2 hours ago
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Differentiation Marathon!
LawofCosine   194
N 3 hours ago by Soupboy0
Hello, everybody!

This is a differentiation marathon. It is just like an ordinary marathon, where you can post problems and provide solutions to the problem posted by the previous user. You can only post differentiation problems (not including integration and differential equations) and please don't make it too hard!

Have fun!

(Sorry about the bad english)
194 replies
LawofCosine
Feb 1, 2025
Soupboy0
3 hours ago
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AoPS Challenge 1
rrusczyk   28
N 3 hours ago by Filipjack
Periodically we'll post difficult challenge problems, which will appear both in this forum and in the forum top bar above.

Here's the first one:

IMAGE
28 replies
rrusczyk
May 23, 2003
Filipjack
3 hours ago
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Turbo the Snail
GreekIdiot   1
N 4 hours ago by GreekIdiot
Let $n$ be a positive integer. There are $n$ circles drawn on a chalkboard such that any two circles intersect at $2$ distinct points and no $3$ circles pass through the same point. Turbo the snail slides along the circles in the following manner, leaving snail goo behind. Initially he moves on one of the circles in clockwise direction. He keeps sliding along until he reaches an intersection with another circle. Then, he continues his journey on this new circle and also changes the direction he is moving in. We define a snail orbit to be the covering of the whole surface of a circle with turbo's goo, and specifically only a single layer of it. Prove that for every odd $n$ there exists at least one configuration of $n$ circles with a single snail orbit, and find all $n$ such that there is exactly one of the aforementioned configuration type.
1 reply
GreekIdiot
Mar 23, 2025
GreekIdiot
4 hours ago
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Additive Combinatorics!
EthanWYX2009   4
N 4 hours ago by GreekIdiot
Source: 2025 TST 15
Let \( X \) be a finite set of real numbers, \( d \) be a real number, and \(\lambda_1, \lambda_2, \cdots, \lambda_{2025}\) be 2025 non-zero real numbers. Define
\[A = \left\{ (x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d \right\},\]\[B = \left\{ (x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0 \right\},\]\[C = \left\{ (x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0 \right\}.\]Show that \( |A|^2 \leq |B| \cdot |C| \).
4 replies
EthanWYX2009
Mar 25, 2025
GreekIdiot
4 hours ago
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A problem
egret_L   0
Mar 18, 2025
A function f from R to R is monotonically increasing if for any x＜y, f(x)＜f(y). Prove that the set of monotonically increasing functions from R into R is equinumerous to R.
0 replies
egret_L
Mar 18, 2025
0 replies
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egret_L
3 posts
egret_L
#1 h  Mar 18, 2025, 5:53 PM
Y by
A function f from R to R is monotonically increasing if for any x＜y, f(x)＜f(y). Prove that the set of monotonically increasing functions from R into R is equinumerous to R.
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