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a My Retirement & New Leadership at AoPS
rrusczyk   1342
N 17 minutes ago by deepanathanv
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
1342 replies
rrusczyk
Monday at 6:37 PM
deepanathanv
17 minutes ago
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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:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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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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n-variable inequality
ABCDE   65
N 10 minutes ago by LMat
Source: 2015 IMO Shortlist A1, Original 2015 IMO #5
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
65 replies
ABCDE
Jul 7, 2016
LMat
10 minutes ago
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2 degree polynomial
PrimeSol   3
N 28 minutes ago by PrimeSol
Let $P_{1}(x)= x^2 +b_{1}x +c_{1}, ... , P_{n}(x)=x^2+ b_{n}x+c_{n}$, $P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$ $\forall i,j ,1 \leq i<j \leq n : P_{i}(x) \ne P_{j}(x)$.
$\forall i,j, 1\leq i<j \leq n : Q_{i,j}(x)= P_{i}(x) + P_{j}(x)$ polynomial with only one root.
$max(n)=?$
3 replies
PrimeSol
Monday at 6:13 AM
PrimeSol
28 minutes ago
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Additive Combinatorics!
EthanWYX2009   3
N 30 minutes ago by flower417477
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 \( |A|^2 \leq |B| \cdot |C| \).
3 replies
EthanWYX2009
Yesterday at 12:49 AM
flower417477
30 minutes ago
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Inspired by IMO 1984
sqing   0
an hour ago
Source: Own
Let $ a,b,c\geq 0 $ and $a^2+b^2+ ab +24abc\geq\frac{81}{64}$. Prove that
$$a+b+\frac{9}{5}c\geq\frac{9}{8}$$$$a+b+\frac{3}{2}c\geq \frac{9}{8}\sqrt [3]{\frac{3}{2}}-\frac{3}{16}$$$$a+b+\frac{8}{5}c\geq \frac{9\sqrt [3]{25}-4}{20}$$Let $ a,b,c\geq 0 $ and $ a^2+b^2+ ab +18abc\geq\frac{343}{324} $. Prove that
$$a+b+\frac{6}{5}c\geq\frac{7\sqrt 7}{18}$$$$a+b+\frac{27}{25}c\geq\frac{35\sqrt [3]5-9}{50}$$
0 replies
sqing
an hour ago
0 replies
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equal angles
jhz   2
N an hour ago by YaoAOPS
Source: 2025 CTST P16
In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$
2 replies
jhz
4 hours ago
YaoAOPS
an hour ago
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Flee Jumping on Number Line
utkarshgupta   23
N an hour ago by Ilikeminecraft
Source: All Russian Olympiad 2015 11.5
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
23 replies
utkarshgupta
Dec 11, 2015
Ilikeminecraft
an hour ago
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Smallest value of |253^m - 40^n|
MS_Kekas   3
N an hour ago by imagien_bad
Source: Kyiv City MO 2024 Round 1, Problem 9.5
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.

Proposed by Oleksii Masalitin
3 replies
MS_Kekas
Jan 28, 2024
imagien_bad
an hour ago
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Operating on lamps in a circle
anantmudgal09   7
N an hour ago by hectorleo123
Source: India Practice TST 2017 D2 P3
There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either on or off. Every second, each lamp undergoes a change according to the following rule:

(a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is off right now. (Indices taken mod $n$.)

(b) Otherwise, $L_i$ is on right now.

Initially, all the lamps are off, except for $L_1$ which is on. Prove that for infinitely many integers $n$ all the lamps will be off eventually, after a finite amount of time.
7 replies
anantmudgal09
Dec 9, 2017
hectorleo123
an hour ago
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2025 Caucasus MO Seniors P1
BR1F1SZ   3
N an hour ago by Mathdreams
Source: Caucasus MO
For given positive integers $a$ and $b$, let us consider the equation$$a + \gcd(b, x) = b + \gcd(a, x).$$[list=a]
[*]For $a = 20$ and $b = 25$, find the least positive integer $x$ satisfying this equation.
[*]Prove that for any positive integers $a$ and $b$, there exist infinitely many positive integers $x$ satisfying this equation.
[/list]
(Here, $\gcd(m, n)$ denotes the greatest common divisor of positive integers $m$ and $n$.)
3 replies
BR1F1SZ
4 hours ago
Mathdreams
an hour ago
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IMO 2018 Problem 2
juckter   95
N an hour ago by Marcus_Zhang
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$for $i = 1, 2, \dots, n$.

Proposed by Patrik Bak, Slovakia
95 replies
juckter
Jul 9, 2018
Marcus_Zhang
an hour ago
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Long condition for the beginning
wassupevery1   2
N 2 hours ago by wassupevery1
Source: 2025 Vietnam IMO TST - Problem 1
Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$holds for all positive rational numbers $x, y$.
2 replies
wassupevery1
Yesterday at 1:49 PM
wassupevery1
2 hours ago
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Inspired by IMO 1984
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +24abc\leq\frac{81}{64}$$Equality holds when $a=b=\frac{3}{8},c=\frac{1}{4}.$
$$a^2+b^2+ ab +18abc\leq\frac{343}{324}$$Equality holds when $a=b=\frac{7}{18},c=\frac{2}{9}.$
0 replies
sqing
2 hours ago
0 replies
J
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Prime-related integers [CMO 2018 - P3]
Amir Hossein   15
N 2 hours ago by Ilikeminecraft
Source: 2018 Canadian Mathematical Olympiad - P3
Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related.

Note that $1$ and $n$ are included as divisors.
15 replies
Amir Hossein
Mar 31, 2018
Ilikeminecraft
2 hours ago
J
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Inspired by IMO 1984
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +17abc\leq\frac{8000}{7803}$$$$a^2+b^2+ ab +\frac{163}{10}abc\leq\frac{7189057}{7173630}$$$$a^2+b^2+ ab +16.23442238abc\le1$$
2 replies
sqing
Yesterday at 3:04 PM
sqing
2 hours ago
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Hyperbolic tangent
MetaphysicalWukong   6
N Mar 23, 2025 by MetaphysicalWukong
Source: Kunwei Ding
Consider the function below. Can someone explain to me why that is the correct answer and find the value of a?
6 replies
MetaphysicalWukong
Mar 23, 2025
MetaphysicalWukong
Mar 23, 2025
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Hyperbolic tangent
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Reveal topic tags
function
algebra
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G H BBookmark kLocked kLocked NReply
Source: Kunwei Ding
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MetaphysicalWukong
246 posts
MetaphysicalWukong
#1 h  Mar 23, 2025, 5:59 AM
Y by
Consider the function below. Can someone explain to me why that is the correct answer and find the value of a?
Attachments:
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aidan0626
1781 posts
aidan0626
#2 h  Mar 23, 2025, 6:02 AM
Y by
Hint: What is the domain of $\tanh^{-1}$?
Z K Y
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MetaphysicalWukong
246 posts
MetaphysicalWukong
#3 h  Mar 23, 2025, 6:12 AM
Y by
aidan0626 wrote:
Hint: What is the domain of $\tanh^{-1}$?

(0,1)
Z K Y
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aidan0626
1781 posts
aidan0626
#4 h  Mar 23, 2025, 6:13 AM
Y by
MetaphysicalWukong wrote:
aidan0626 wrote:
Hint: What is the domain of $\tanh^{-1}$?

(0,1)

you sure?
Z K Y
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MetaphysicalWukong
246 posts
MetaphysicalWukong
#5 h  Mar 23, 2025, 6:15 AM
Y by
aidan0626 wrote:
MetaphysicalWukong wrote:
aidan0626 wrote:
Hint: What is the domain of $\tanh^{-1}$?

(0,1)

you sure?

(-1,1)
Z K Y
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aidan0626
1781 posts
aidan0626
#6 h  Mar 23, 2025, 6:16 AM
Y by
well you have your answer then right
Z K Y
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MetaphysicalWukong
246 posts
MetaphysicalWukong
#7 h  Mar 23, 2025, 6:18 AM
Y by
aidan0626 wrote:
well you have your answer then right

a is equal to one. am i right? also how to remember the domains of hyperbolic trig functions?
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