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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
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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IMO ShortList 1998, number theory problem 6
orl   28
N 14 minutes ago by Zany9998
Source: IMO ShortList 1998, number theory problem 6
For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.
28 replies
orl
Oct 22, 2004
Zany9998
14 minutes ago
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A projectional vision in IGO
Shayan-TayefehIR   14
N 19 minutes ago by mathuz
Source: IGO 2024 Advanced Level - Problem 3
In the triangle $\bigtriangleup ABC$ let $D$ be the foot of the altitude from $A$ to the side $BC$ and $I$, $I_A$, $I_C$ be the incenter, $A$-excenter, and $C$-excenter, respectively. Denote by $P\neq B$ and $Q\neq D$ the other intersection points of the circle $\bigtriangleup BDI_C$ with the lines $BI$ and $DI_A$, respectively. Prove that $AP=AQ$.

Proposed Michal Jan'ik - Czech Republic
14 replies
Shayan-TayefehIR
Nov 14, 2024
mathuz
19 minutes ago
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(a²-b²)(b²-c²) = abc
straight   3
N 21 minutes ago by straight
Find all triples of positive integers $(a,b,c)$ such that

\[(a^2-b^2)(b^2-c^2) = abc.\]
If you can't solve this, assume $gcd(a,c) = 1$. If this is still too hard assume in $a \ge b \ge c$ that $b-c$ is a prime.
3 replies
straight
Mar 24, 2025
straight
21 minutes ago
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A checkered square consists of dominos
nAalniaOMliO   1
N 22 minutes ago by BR1F1SZ
Source: Belarusian National Olympiad 2025
A checkered square $8 \times 8$ is divided into rectangles with two cells. Two rectangles are called adjacent if they share a segment of length 1 or 2. In each rectangle the amount of adjacent with it rectangles is written.
Find the maximal possible value of the sum of all numbers in rectangles.
1 reply
nAalniaOMliO
Yesterday at 8:21 PM
BR1F1SZ
22 minutes ago
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A lot of numbers and statements
nAalniaOMliO   2
N an hour ago by nAalniaOMIiO
Source: Belarusian National Olympiad 2025
101 numbers are written in a circle. Near the first number the statement "This number is bigger than the next one" is written, near the second "This number is bigger that the next two" and etc, near the 100th "This number is bigger than the next 100 numbers".
What is the maximum possible amount of the statements that can be true?
2 replies
1 viewing
nAalniaOMliO
Yesterday at 8:20 PM
nAalniaOMIiO
an hour ago
J
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USAMO 1981 #2
Mrdavid445   9
N an hour ago by Marcus_Zhang
Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number of communities in this county.
9 replies
Mrdavid445
Jul 26, 2011
Marcus_Zhang
an hour ago
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Monkeys have bananas
nAalniaOMliO   2
N an hour ago by nAalniaOMIiO
Source: Belarusian National Olympiad 2025
Ten monkeys have 60 bananas. Each monkey has at least one banana and any two monkeys have different amounts of bananas.
Prove that any six monkeys can distribute their bananas between others such that all 4 remaining monkeys have the same amount of bananas.
2 replies
nAalniaOMliO
Yesterday at 8:20 PM
nAalniaOMIiO
an hour ago
J
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A number theory problem from the British Math Olympiad
Rainbow1971   12
N 2 hours ago by ektorasmiliotis
Source: British Math Olympiad, 2006/2007, round 1, problem 6
I am a little surprised to find that I am (so far) unable to solve this little problem:

[quote]Let $n$ be an integer. Show that, if $2 + 2 \sqrt{1+12n^2}$ is an integer, then it is a perfect square.[/quote]

I set $k := \sqrt{1+12n^2}$. If $2 + 2 \sqrt{1+12n^2}$ is an integer, then $k (=\sqrt{1+12n^2})$ is at least rational, so that $1 + 12n^2$ must be a perfect square then. Using Conway's topograph method, I have found out that the smallest non-negative pairs $(n, k)$ for which this happens are $(0,1), (2,7), (28,97)$ and $(390, 1351)$, and that, for every such pair $(n,k)$, the "next" such pair can be calculated as
$$ \begin{bmatrix} 7 & 2 \\ 24 & 7 \end{bmatrix} \begin{bmatrix} n \\ k \end{bmatrix} .$$The eigenvalues of that matrix are irrational, however, so that any calculation which uses powers of that matrix is a little cumbersome. There must be an easier way, but I cannot find it. Can you?

Thank you.




12 replies
1 viewing
Rainbow1971
Yesterday at 8:39 PM
ektorasmiliotis
2 hours ago
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A number theory about divisors which no one fully solved at the contest
nAalniaOMliO   22
N 2 hours ago by nAalniaOMliO
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
22 replies
nAalniaOMliO
Jul 24, 2024
nAalniaOMliO
2 hours ago
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D1018 : Can you do that ?
Dattier   1
N 2 hours ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
2 hours ago
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Nordic 2025 P3
anirbanbz   8
N 3 hours ago by Primeniyazidayi
Source: Nordic 2025
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$.
8 replies
anirbanbz
Mar 25, 2025
Primeniyazidayi
3 hours ago
J
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f( - f (x) - f (y))= 1 -x - y , in Zxz
parmenides51   6
N 3 hours ago by Chikara
Source: 2020 Dutch IMO TST 3.3
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$for all $x, y \in Z$
6 replies
parmenides51
Nov 22, 2020
Chikara
3 hours ago
J
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Hard limits
Snoop76   2
N 3 hours ago by maths_enthusiast_0001
$a_n$ and $b_n$ satisfies the following recursion formulas: $a_{0}=1, $ $b_{0}=1$, $ a_{n+1}=a_{n}+b_{n}$$ $ and $ $$ b_{n+1}=(2n+3)b_{n}+a_{n}$. Find $ \lim_{n \to \infty} \frac{a_n}{(2n-1)!!}$ $ $ and $ $ $\lim_{n \to \infty} \frac{b_n}{(2n+1)!!}.$
2 replies
Snoop76
Mar 25, 2025
maths_enthusiast_0001
3 hours ago
J
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2025 Caucasus MO Seniors P2
BR1F1SZ   3
N 4 hours ago by X.Luser
Source: Caucasus MO
Let $ABC$ be a triangle, and let $B_1$ and $B_2$ be points on segment $AC$ symmetric with respect to the midpoint of $AC$. Let $\gamma_A$ denote the circle passing through $B_1$ and tangent to line $AB$ at $A$. Similarly, let $\gamma_C$ denote the circle passing through $B_1$ and tangent to line $BC$ at $C$. Let the circles $\gamma_A$ and $\gamma_C$ intersect again at point $B'$ ($B' \neq B_1$). Prove that $\angle ABB' = \angle CBB_2$.
3 replies
BR1F1SZ
Mar 26, 2025
X.Luser
4 hours ago
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Inspired by m4thbl3nd3r
sqing   1
N Mar 27, 2025 by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=3$. Prove that$$a^3b+b^3c+c^3a+\frac{1419}{256}abc\le\frac{2187}{256}$$Equality holds when $ a=b=c=1 $ or $ a=0,b=\frac{9}{4},c=\frac{3}{4} $ or $ a=\frac{3}{4} ,b=0,c=\frac{9}{4} $
or $ a=\frac{9}{4} ,b=\frac{3}{4},c=0. $
1 reply
sqing
Mar 22, 2025
sqing
Mar 27, 2025
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Inspired by m4thbl3nd3r
G H J
Reveal topic tags
inequalities proposed
algebra
inequalities
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Source: Own
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sqing
41310 posts
sqing
#1 h  Mar 22, 2025, 3:35 PM
Y by
Let $ a,b,c\geq 0 $ and $a+b+c=3$. Prove that$$a^3b+b^3c+c^3a+\frac{1419}{256}abc\le\frac{2187}{256}$$Equality holds when $ a=b=c=1 $ or $ a=0,b=\frac{9}{4},c=\frac{3}{4} $ or $ a=\frac{3}{4} ,b=0,c=\frac{9}{4} $
or $ a=\frac{9}{4} ,b=\frac{3}{4},c=0. $
This post has been edited 2 times. Last edited by sqing, Mar 22, 2025, 3:40 PM
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The post below has been deleted. Click to close.
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sqing
41310 posts
sqing
#2 h  Mar 27, 2025, 1:22 PM
Y by
Let $ a,b,c\geq 0 $ and $ a^2+b^2+c^2+ab+bc+ca=6. $ Prove that$$a^3b+b^3c+c^3a \le\frac{13\sqrt{13}-35}{2}$$
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