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k a My Retirement & New Leadership at AoPS
rrusczyk   1571
N Mar 26, 2025 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
+2 w
rrusczyk
Mar 24, 2025
SmartGroot
Mar 26, 2025
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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]
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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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Polynomials
Pao_de_sal   2
N 36 minutes ago by ektorasmiliotis
find all natural numbers n such that the polynomial x²ⁿ + xⁿ + 1 is divisible by x² + x + 1
2 replies
+2 w
Pao_de_sal
an hour ago
ektorasmiliotis
36 minutes ago
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April Fools Geometry
awesomeming327.   2
N an hour ago by avinashp
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ be the projection from $A$ onto $BC$. Let $E$ be a point on the extension of $AD$ past $D$ such that $\angle BAC+\angle BEC=90^\circ$. Let $L$ be on the perpendicular bisector of $AE$ such that $L$ and $C$ are on the same side of $AE$ and
\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]Let the reflection of $D$ across $AB$ and $AC$ be $W$ and $Y$, respectively. Let $X\in AW$ and $Z\in AY$ such that $\angle XBE=\angle ZCE=90^\circ$. Let $EX$ and $EZ$ intersect the circumcircles of $EBD$ and $ECD$ at $J$ and $K$, respectively. Let $LB$ and $LC$ intersect $WJ$ and $YK$ at $P$ and $Q$. Let $PQ$ intersect $BC$ at $F$. Prove that $FB/FC=DB/DC$.
2 replies
awesomeming327.
Today at 2:52 PM
avinashp
an hour ago
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inequalities
Cobedangiu   2
N an hour ago by ehuseyinyigit
Source: own
$a,b>0$ and $a+b=1$. Find min P:
$P=\sqrt{\frac{1-a}{1+7a}}+\sqrt{\frac{1-b}{1+7b}}$
2 replies
Cobedangiu
3 hours ago
ehuseyinyigit
an hour ago
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very cute geo
rafaello   3
N an hour ago by bin_sherlo
Source: MODSMO 2021 July Contest P7
Consider a triangle $ABC$ with incircle $\omega$. Let $S$ be the point on $\omega$ such that the circumcircle of $BSC$ is tangent to $\omega$ and let the $A$-excircle be tangent to $BC$ at $A_1$. Prove that the tangent from $S$ to $\omega$ and the tangent from $A_1$ to $\omega$ (distinct from $BC$) meet on the line parallel to $BC$ and passing through $A$.
3 replies
rafaello
Oct 26, 2021
bin_sherlo
an hour ago
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f(n+1) = f(n) + 2^f(n) implies f(n) distinct mod 3^2013
v_Enhance   50
N an hour ago by Maximilian113
Source: USA TSTST 2013, Problem 8
Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$, $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.
50 replies
v_Enhance
Aug 13, 2013
Maximilian113
an hour ago
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Ez Number Theory
IndoMathXdZ   40
N 2 hours ago by akliu
Source: IMO SL 2018 N1
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
40 replies
IndoMathXdZ
Jul 17, 2019
akliu
2 hours ago
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Is this FE solvable?
Mathdreams   0
3 hours ago
Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]for all reals $x$ and $y$.
0 replies
Mathdreams
3 hours ago
0 replies
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OFM2021 Senior P1
medhimdi   0
3 hours ago
Let $a_1, a_2, a_3, \dots$ and $b_1, b_2, b_3, \dots$ be two sequences of integers such that $a_{n+2}=a_{n+1}+a_n$ and $b_{n+2}=b_{n+1}+b_n$ for all $n\geq1$. Suppose that $a_n$ divides $b_n$ for an infinity of integers $n\geq1$. Prove that there exist an integer $c$ such that $b_n=ca_n$ for all $n\geq1$
0 replies
medhimdi
3 hours ago
0 replies
J
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Hard NT problem
tiendat004   2
N 3 hours ago by avinashp
Given two odd positive integers $a,b$ are coprime. Consider the sequence $(x_n)$ given by $x_0=2,x_1=a,x_{n+2}=ax_{n+1}+bx_n,$ $\forall n\geq 0$. Suppose that there exist positive integers $m,n,p$ such that $mnp$ is even and $\dfrac{x_m}{x_nx_p}$ is an integer. Prove that the numerator in its simplest form of $\dfrac{m}{np}$ is an odd integer greater than $1$.
2 replies
tiendat004
Aug 15, 2024
avinashp
3 hours ago
J
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disjoint subsets
nayel   2
N 3 hours ago by alexanderhamilton124
Source: Taiwan 2001
Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$. Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint.

what i have is this
2 replies
nayel
Apr 18, 2007
alexanderhamilton124
3 hours ago
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Modular Arithmetic and Integers
steven_zhang123   2
N 3 hours ago by GreekIdiot
Integers \( n, a, b \in \mathbb{Z}^+ \) satisfies \( n + a + b = 30 \). If \( \alpha < b, \alpha \in \mathbb{Z^+} \), find the maximum possible value of $\sum_{k=1}^{\alpha} \left \lfloor \frac{kn^2 \bmod a }{b-k} \right \rfloor $.
2 replies
steven_zhang123
Mar 28, 2025
GreekIdiot
3 hours ago
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f(x+y)f(z)=f(xz)+f(yz)
dangerousliri   30
N 3 hours ago by GreekIdiot
Source: Own
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all irrational numbers $x, y$ and $z$,
$$f(x+y)f(z)=f(xz)+f(yz)$$
Some stories about this problem. This problem it is proposed by me (Dorlir Ahmeti) and Valmir Krasniqi. We did proposed this problem for IMO twice, on 2018 and on 2019 from Kosovo. None of these years it wasn't accepted and I was very surprised that it wasn't selected at least for shortlist since I think it has a very good potential. Anyway I hope you will like the problem and you are welcomed to give your thoughts about the problem if it did worth to put on shortlist or not.
30 replies
dangerousliri
Jun 25, 2020
GreekIdiot
3 hours ago
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Unsolved NT, 3rd time posting
GreekIdiot   6
N 3 hours ago by GreekIdiot
Source: own
Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
6 replies
GreekIdiot
Mar 26, 2025
GreekIdiot
3 hours ago
J
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Need hint:''(
Buh_-1235   0
3 hours ago
Source: Canada Winter mock 2015
Recall that for any positive integer m, φ(m) denotes the number of positive integers less than m which are relatively
prime to m. Let n be an odd positive integer such that both φ(n) and φ(n + 1) are powers of two. Prove n + 1 is power
of two or n = 5.
0 replies
Buh_-1235
3 hours ago
0 replies
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J
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You just need to throw facts
vicentev   3
N Mar 29, 2025 by MathSaiyan
Source: TST IMO CHILE 2025
Let \( a, b, c, d \) be real numbers such that \( abcd = 1 \), and
\[ a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0. \]Prove that one of the numbers \( ab, ac \) or \( ad \) is equal to \( -1 \).
3 replies
vicentev
Mar 29, 2025
MathSaiyan
Mar 29, 2025
J
You just need to throw facts
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Reveal topic tags
algebra
Chile
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G H BBookmark kLocked kLocked NReply
Source: TST IMO CHILE 2025
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vicentev
136 posts
vicentev
#1 h  Mar 29, 2025, 3:25 PM
Y by
Let \( a, b, c, d \) be real numbers such that \( abcd = 1 \), and
\[ a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0. \]Prove that one of the numbers \( ab, ac \) or \( ad \) is equal to \( -1 \).
Z K Y
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LolMathZ13PR
2 posts
LolMathZ13PR
#2 h  Mar 29, 2025, 3:33 PM
Y by
si usas logica y razonamiento te daras cuenta que obligatoriamente ab,ac,ad deben ser un -1 si no fuera asi no cumpliria con abcd=1 y por lo tanto contradiccion y probado pero a los de las competencias no les gusta esto ARRIBA RAMANUJAN!!!!!!!!
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E50
5 posts
E50
#3 h  Mar 29, 2025, 3:42 PM
Y by
$a+b+c+d+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d} = 0$ $ \Longrightarrow$ $a+b+c+d+bcd+\frac{1}{b}+\frac{1}{c}+\frac{1}{d} = 0$ $ \Longrightarrow$ $a^2+\frac{a}{b}+\frac{a}{c}+\frac{a}{d}+ab+ac+ad+1 = 0$ $ \Longrightarrow$ $a^3bcd+a^2bc+a^2bd+a^2cd+ab+ac+ad+1 = 0$ $ \Longrightarrow$ $(ab+1)(ac+1)(ad+1) = 0$. Thus, one of the numbers $ab,ac$ or $ad$ is equal to $-1$.
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MathSaiyan
74 posts
MathSaiyan
#4 h  Mar 29, 2025, 3:46 PM
Y by
Old, see https://artofproblemsolving.com/community/c6h3278304p30182928
This post has been edited 1 time. Last edited by MathSaiyan, Mar 29, 2025, 3:46 PM
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