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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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Numbers not power of 5
Kayak   33
N 5 minutes ago by ihategeo_1969
Source: Indian TST D1 P2
Show that there do not exist natural numbers $a_1, a_2, \dots, a_{2018}$ such that the numbers \[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]are all powers of $5$

Proposed by Tejaswi Navilarekallu
33 replies
Kayak
Jul 17, 2019
ihategeo_1969
5 minutes ago
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Chile TST IMO prime geo
vicentev   4
N 6 minutes ago by Retemoeg
Source: TST IMO CHILE 2025
Let \( ABC \) be a triangle with \( AB < AC \). Let \( M \) be the midpoint of \( AC \), and let \( D \) be a point on segment \( AC \) such that \( DB = DC \). Let \( E \) be the point of intersection, different from \( B \), of the circumcircle of triangle \( ABM \) and line \( BD \). Define \( P \) and \( Q \) as the points of intersection of line \( BC \) with \( EM \) and \( AE \), respectively. Prove that \( P \) is the midpoint of \( BQ \).
4 replies
vicentev
Today at 2:35 AM
Retemoeg
6 minutes ago
J
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Cute orthocenter geometry
MarkBcc168   77
N 9 minutes ago by ErTeeEs06
Source: ELMO 2020 P4
Let acute scalene triangle $ABC$ have orthocenter $H$ and altitude $AD$ with $D$ on side $BC$. Let $M$ be the midpoint of side $BC$, and let $D'$ be the reflection of $D$ over $M$. Let $P$ be a point on line $D'H$ such that lines $AP$ and $BC$ are parallel, and let the circumcircles of $\triangle AHP$ and $\triangle BHC$ meet again at $G \neq H$. Prove that $\angle MHG = 90^\circ$.

Proposed by Daniel Hu.
77 replies
MarkBcc168
Jul 28, 2020
ErTeeEs06
9 minutes ago
J
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A functional equation from MEMO
square_root_of_3   23
N 15 minutes ago by John_Mgr
Source: Middle European Mathematical Olympiad 2022, problem I-1
Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(x+f(x+y))=x+f(f(x)+y)$$holds for all real numbers $x$ and $y$.
23 replies
square_root_of_3
Sep 1, 2022
John_Mgr
15 minutes ago
J
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A weird inequality
Eeightqx   0
22 minutes ago
For all $a,\,b,\,c>0$, find the maximum $\lambda$ which satisfies
$$\sum_{cyc}a^2(a-2b)(a-\lambda b)\ge 0.$$hint
0 replies
Eeightqx
22 minutes ago
0 replies
J
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Student's domination
Entei   0
27 minutes ago
Given $n$ students and their test results on $k$ different subjects, we say that student $A$ dominates student $B$ if and only if $A$ outperforms $B$ on all subjects. Assume that no two of them have the same score on the same subject, find the probability that there exists a pair of domination in class.
0 replies
Entei
27 minutes ago
0 replies
J
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The Curious Equation for ConoSur
vicentev   3
N 33 minutes ago by AshAuktober
Source: TST IMO-CONO CHILE 2025
Find all triples \( (x, y, z) \) of positive integers that satisfy the equation
\[ x + xy + xyz = 31. \]
3 replies
vicentev
an hour ago
AshAuktober
33 minutes ago
J
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You just need to throw facts
vicentev   3
N 34 minutes ago 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
an hour ago
MathSaiyan
34 minutes ago
J
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A number theory problem from the British Math Olympiad
Rainbow1971   5
N 35 minutes ago by Rainbow1971
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.




5 replies
Rainbow1971
Yesterday at 8:39 PM
Rainbow1971
35 minutes ago
J
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Finding maximum sum of consecutive ten numbers in circle.
Goutham   3
N an hour ago by FarrukhKhayitboyev
Each of $999$ numbers placed in a circular way is either $1$ or $-1$. (Both values appear). Consider the total sum of the products of every $10$ consecutive numbers.
$(a)$ Find the minimal possible value of this sum.
$(b)$ Find the maximal possible value of this sum.
3 replies
Goutham
Feb 8, 2011
FarrukhKhayitboyev
an hour ago
J
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The Chile Awkward Party
vicentev   0
an hour ago
Source: TST IMO CHILE 2025
At a meeting, there are \( N \) people who do not know each other. Prove that it is possible to introduce them in such a way that no three of them have the same number of acquaintances.
0 replies
vicentev
an hour ago
0 replies
J
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Sharygin CR P20
TheDarkPrince   37
N an hour ago by E50
Source: Sharygin 2018
Let the incircle of a nonisosceles triangle $ABC$ touch $AB$, $AC$ and $BC$ at points $D$, $E$ and $F$ respectively. The corresponding excircle touches the side $BC$ at point $N$. Let $T$ be the common point of $AN$ and the incircle, closest to $N$, and $K$ be the common point of $DE$ and $FT$. Prove that $AK||BC$.
37 replies
TheDarkPrince
Apr 4, 2018
E50
an hour ago
J
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Fibonacci sequence and primes
vicentev   0
an hour ago
Source: TST IMO CHILE 2025
Let \( u_n \) be the \( n \)-th term of the Fibonacci sequence (where \( u_1 = u_2 = 1 \) and \( u_{n+1} = u_n + u_{n-1} \) for \( n \geq 2 \)). For each prime \( p \), let \( n(p) \) be the smallest integer \( n \) such that \( u_n \) is divisible by \( p \). Find the smallest possible value of \( p - n(p) \).
0 replies
vicentev
an hour ago
0 replies
J
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Gheorghe Țițeica 2025 Grade 7 P3
AndreiVila   1
N an hour ago by Rainbow1971
Source: Gheorghe Țițeica 2025
Out of all the nondegenerate triangles with positive integer sides and perimeter $100$, find the one with the smallest area.
1 reply
AndreiVila
Yesterday at 8:41 PM
Rainbow1971
an hour ago
J
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J
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Thanks u!
Ruji2018252   4
N Mar 26, 2025 by pco
Find all $f:\mathbb{R}\to\mathbb{R}$ and
\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]
4 replies
Ruji2018252
Mar 26, 2025
pco
Mar 26, 2025
J
Thanks u!
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Reveal topic tags
functional equation
IMO
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G H BBookmark kLocked kLocked NReply
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Ruji2018252
377 posts
Ruji2018252
#1 h  Mar 26, 2025, 8:45 AM
Y by
Find all $f:\mathbb{R}\to\mathbb{R}$ and
\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]
This post has been edited 1 time. Last edited by Ruji2018252, Mar 26, 2025, 9:30 AM
Reason: Sori
Z K Y
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Mhremath
67 posts
Mhremath
#2 h  Mar 26, 2025, 9:05 AM • 1 Y
Y by pco
$P(0,0)\implies f(0)=0$
$P(0,x)\implies f(f(x))=x$
$P(x,0)\implies f(x^2)=x$ for all x
but it leads contradiction
Z K Y
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Haris1
66 posts
Haris1
#3 h  Mar 26, 2025, 9:16 AM
Y by
If u can solve this then u can also do $IMO$ $2017$ $P2$.
Z K Y
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Ruji2018252
377 posts
Ruji2018252
#4 h  Mar 26, 2025, 9:30 AM
Y by
Mhremath wrote:
$P(0,0)\implies f(0)=0$
$P(0,x)\implies f(f(x))=x$
$P(x,0)\implies f(x^2)=x$ for all x
but it leads contradiction

Me update new problem bro!
Z K Y
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pco
23488 posts
pco
#6 h  Mar 26, 2025, 11:25 AM • 1 Y
Y by MS_asdfgzxcvb
Ruji2018252 wrote:
Find all $f:\mathbb{R}\to\mathbb{R}$ and
\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]
Let $P(x,y)$ be the assertion $f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y$
Let $a=f(0)$

$P(0,x)$ $\implies$ $f(f(x))=x+c$ where $c=f(a)^2+a$ and so $f(x)$ is bijective
$P(x,y)$ may be written as new assertion $Q(x,y)$ : $f(x+y)+f(x^2+f(y))=(x+c)^2+f(x)+f(y)+y$

1) $c=0$ and so $f(f(x))=x$
Proof

2) $\boxed{f(x)=x\quad\forall x}$
Proof
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