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a My Retirement & New Leadership at AoPS
rrusczyk   933
N a few seconds ago by Dolphincurious79
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
933 replies
rrusczyk
Yesterday at 6:37 PM
Dolphincurious79
a few seconds 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:
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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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Maximum of Incenter-triangle
mpcnotnpc   0
41 minutes ago
Triangle $\Delta ABC$ has side lengths $a$, $b$, and $c$. Select a point $P$ inside $\Delta ABC$, and construct the incenters of $\Delta PAB$, $\Delta PBC$, and $\Delta PAC$ and denote them as $I_A$, $I_B$, $I_C$. What is the maximum area of the triangle $\Delta I_A I_B I_C$?
0 replies
mpcnotnpc
41 minutes ago
0 replies
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polynomial and number theory
srnjbr   0
44 minutes ago
n is a natural number and f is a polynomial with integer coefficients. We know that for every m integer members, there is an integer k such that f(k)-m is divisible by n. Show that there is a polynomial g(x) such that f(g(m))-m is divisible by n.
0 replies
srnjbr
44 minutes ago
0 replies
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2019 Polynomial problem
srnjbr   0
an hour ago
suppose t is a member of the interval (1,2). show that there exists a polynomial p with coefficients +-1 such that |p(t)-2019|<=1
0 replies
srnjbr
an hour ago
0 replies
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Product of f(m) multiple of odd integers
buzzychaoz   23
N 2 hours ago by john0512
Source: China Team Selection Test 2016 Test 2 Day 2 Q4
Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.
23 replies
buzzychaoz
Mar 21, 2016
john0512
2 hours ago
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God do bosses have a hard job
AshAuktober   1
N 2 hours ago by tyrantfire4
Source: 05JPN5 (couldn't find on search function)
The boss has to assign ten job positions to ten candidates, considering two parameters: preference and ability. If candidate A prefers job $v$ to job $u$ and has a better ability in job $v$ than candidate B, but A is assigned job $u$ and B is assigned job $v$, then A will complain. Also, if it is possible to assign each job to a candidate with a higher ability, the director will complain. Show that the boss can assign the jobs so as to avoid any complaints.
1 reply
AshAuktober
2 hours ago
tyrantfire4
2 hours ago
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Nordic 2025 P2
anirbanbz   5
N 2 hours ago by KAME06
Source: Nordic 2025
Let $p$ be a prime and suppose $2^{2p} \equiv 1 (\text{mod}$ $ 2p+1)$ is prime. Prove that $2p+1$ is prime$^{1}$

$^{1}$This is a special case of Pocklington's theorem. A proof of this special case is required.
5 replies
anirbanbz
Today at 12:35 PM
KAME06
2 hours ago
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Nice problem
hanzo.ei   6
N 2 hours ago by Ianis

Given two sequences $(a_n)$ and $(b_n)$ satisfying $(a_n + b_n)a_n \neq 0$ for all $n$, and both series
\[ \sum \frac{a_n}{b_n}, \quad \sum \frac{b_n}{a_n} \]are convergent. Prove that the series
\[ \sum \frac{a_n}{a_n + b_n} \]also converges.
6 replies
hanzo.ei
Today at 12:12 PM
Ianis
2 hours ago
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System of Equations in Z
rightways   0
2 hours ago
Source: Kazakhstan NMO Problem 1
Find all quadruples $a,b,c,d$ for which $2^a+3^b=5^c d$ and $2^b+3^a=5^d c$.
0 replies
rightways
2 hours ago
0 replies
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$x^{y^2+1}+y^{x^2+1}=2^z$
Zahy2106   0
3 hours ago
Source: Collection
Find all $(x,y,z)\in (\mathbb{Z^+})^3$ safisty: $x^{y^2+1}+y^{x^2+1}=2^z$
0 replies
Zahy2106
3 hours ago
0 replies
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Solve this:
slimshady360   1
N 3 hours ago by pco
Solve this:
1 reply
slimshady360
5 hours ago
pco
3 hours ago
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Nordic 2025 P1
anirbanbz   4
N 4 hours ago by Mathdreams
Source: Nordic 2025
Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying:
$(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$
4 replies
anirbanbz
Today at 12:32 PM
Mathdreams
4 hours ago
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Slightly weird points which are not so weird
Pranav1056   11
N 4 hours ago by maths_enthusiast_0001
Source: India TST 2023 Day 4 P1
Suppose an acute scalene triangle $ABC$ has incentre $I$ and incircle touching $BC$ at $D$. Let $Z$ be the antipode of $A$ in the circumcircle of $ABC$. Point $L$ is chosen on the internal angle bisector of $\angle BZC$ such that $AL = LI$. Let $M$ be the midpoint of arc $BZC$, and let $V$ be the midpoint of $ID$. Prove that $\angle IML = \angle DVM$
11 replies
Pranav1056
Jul 9, 2023
maths_enthusiast_0001
4 hours ago
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Inequality involving a unimodal sequence
timon92   1
N 4 hours ago by math_comb01
Source: 76 Polish MO, 2nd round, p6
Let $1\le k\le n$. Suppose that the sequence $a_1, a_2, \ldots, a_n$ satisfies $0\le a_1 \le a_2 \le \ldots \le a_k$ and $0 \le a_n \le a_{n-1} \le \ldots \le a_k$. The sequence $b_1, b_2, \ldots, b_n$ is the nondecreasing permutation of $a_1, a_2, \ldots, a_n$. Prove that
\[\sum_{i=1}^n \sum_{j=1}^n (j-i)^2a_ia_j \le \sum_{i=1}^n \sum_{j=1}^n (j-i)^2b_ib_j \]
1 reply
1 viewing
timon92
Feb 15, 2025
math_comb01
4 hours ago
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Decomposition into bounded number of factors
fattypiggy123   11
N 4 hours ago by sttsmet
Source: China TST 3 Day 1 Q3
Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.
11 replies
fattypiggy123
Mar 23, 2015
sttsmet
4 hours ago
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2 var inquality
sqing   3
N Mar 23, 2025 by sqing
Source: Own
Let $ a,b>0 $ and $ 3a+4b=a^3b^2. $ Prove that
$$2a+b+\dfrac{2}{a}+\dfrac{3}{b}\geq \frac{11}{\sqrt2}$$$$a+\dfrac{2}{a}+\dfrac{3}{b}\geq 4\sqrt[4]{\frac23}$$$$\dfrac{2}{a}+\dfrac{3}{b}\geq 2\sqrt[4]3$$$$3a+\dfrac{2}{a}+\dfrac{3}{b}\geq \sqrt[4]{354+66\sqrt{33}}$$
3 replies
sqing
Mar 4, 2025
sqing
Mar 23, 2025
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2 var inquality
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Reveal topic tags
inequalities proposed
inequalities
algebra
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Source: Own
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sqing
41233 posts
sqing
#1 h  Mar 4, 2025, 12:46 PM
Y by
Let $ a,b>0 $ and $ 3a+4b=a^3b^2. $ Prove that
$$2a+b+\dfrac{2}{a}+\dfrac{3}{b}\geq \frac{11}{\sqrt2}$$$$a+\dfrac{2}{a}+\dfrac{3}{b}\geq 4\sqrt[4]{\frac23}$$$$\dfrac{2}{a}+\dfrac{3}{b}\geq 2\sqrt[4]3$$$$3a+\dfrac{2}{a}+\dfrac{3}{b}\geq \sqrt[4]{354+66\sqrt{33}}$$
This post has been edited 2 times. Last edited by sqing, Mar 4, 2025, 12:58 PM
Z K Y
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sqing
41233 posts
sqing
#2 h  Mar 10, 2025, 8:38 AM
Y by
Let $ a,b>0 $ and $ \dfrac{1}{a}+\dfrac{2}{b}\leq 2 . $ Prove that
$$ a+b \leq a^3b^2 $$Let $ a,b>0 $ and $ \dfrac{3}{a}+\dfrac{2}{b}\leq 2\sqrt 3 . $ Prove that
$$ a+3b \leq a^3b^2 $$Let $ a,b>0 $ and $ \dfrac{1}{a}+\dfrac{8}{b}\leq 2\sqrt 2 . $ Prove that
$$ 4a+b \leq a^3b^2 $$
Z K Y
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sqing
41233 posts
sqing
#3 h  Mar 19, 2025, 5:00 AM
Y by
Let $ a,b>0 $ and $ 3a+4b=a^3b^2. $ Prove that
$$2a+b+ka^2b +\frac{2}{a}+\frac{3}{b}\geq \frac{6k+11}{\sqrt2}$$Where $ k\geq 0.$
$$2a+b+a^2b+\frac{2}{a}+\frac{3}{b} \geq \frac{17}{\sqrt2}$$
This post has been edited 1 time. Last edited by sqing, Mar 19, 2025, 5:03 AM
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sqing
41233 posts
sqing
#4 h  Mar 23, 2025, 3:07 AM
Y by
sqing wrote:
Let $ a,b>0 $ and $ 3a+4b=a^3b^2. $ Prove that
$$2a+b+\dfrac{2}{a}+\dfrac{3}{b}\geq \frac{11}{\sqrt2}$$
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