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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
1 viewing
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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2023 factors and perfect cube
proxima1681   5
N an hour ago by mqoi_KOLA
Source: Indian Statistical Institute (ISI) UGB 2023 P4
Let $n_1, n_2, \cdots , n_{51}$ be distinct natural numbers each of which has exactly $2023$ positive integer factors. For instance, $2^{2022}$ has exactly $2023$ positive integer factors $1,2, 2^{2}, 2^{3}, \cdots 2^{2021}, 2^{2022}$. Assume that no prime larger than $11$ divides any of the $n_{i}$'s. Show that there must be some perfect cube among the $n_{i}$'s.
5 replies
proxima1681
May 14, 2023
mqoi_KOLA
an hour ago
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Integer Symmetric Polynomials
proxima1681   4
N an hour ago by mqoi_KOLA
Source: Indian Statistical Institute (ISI) UGB 2023 P7
(a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta= XY+YZ+ZX$ and $\gamma = XYZ$.
(b) Let $G_n=x^n \sin(nA)+y^n \sin(nB)+z^n \sin(nC)$, where $x,y,z, A,B,C$ are real numbers such that $A+B+C$ is an integral multiple of $\pi$. Using (a) or otherwise show that if $G_1=G_2=0$, then $G_n=0$ for all positive integers $n$.
4 replies
proxima1681
May 14, 2023
mqoi_KOLA
an hour ago
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IMO 2008, Question 1
orl   153
N an hour ago by QueenArwen
Source: IMO Shortlist 2008, G1
Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$.

Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic.

Author: Andrey Gavrilyuk, Russia
153 replies
orl
Jul 16, 2008
QueenArwen
an hour ago
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P(x) doesn't have 2024 distinct real roots
gatnghiep   1
N an hour ago by kiyoras_2001
Given a polynomial $P(x)=x^{2024}+a_{2023}x^{2023}+...+a_1x+1$ with real coefficient. It is known that $|a_{1012}|<2$ and $a_k = a_{2024-k}, \forall k = 1,2,...,2012$. Prove that $P(x)$ can't have $2024$ distinct real roots.
1 reply
1 viewing
gatnghiep
Oct 2, 2024
kiyoras_2001
an hour ago
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A lies on the radical axis of BQX and CPX
a_507_bc   35
N 2 hours ago by jordiejoh
Source: APMO 2024 P1
Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.
35 replies
a_507_bc
Jul 29, 2024
jordiejoh
2 hours ago
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sum(ab/4a^2+b^2) <= 3/5
truongphatt2668   3
N 2 hours ago by Nguyenhuyen_AG
Source: I remember I read it somewhere
Let $a,b,c>0$. Prove that:
$$\dfrac{ab}{a^2+4b^2} + \dfrac{bc}{b^2+4c^2} + \dfrac{ca}{c^2+4a^2} \le \dfrac{3}{5}$$
3 replies
truongphatt2668
Monday at 1:23 PM
Nguyenhuyen_AG
2 hours ago
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the epitome of olympiad nt
youlost_thegame_1434   30
N 2 hours ago by Jupiterballs
Source: 2023 IMO Shortlist N3
For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.
30 replies
youlost_thegame_1434
Jul 17, 2024
Jupiterballs
2 hours ago
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inequalities
Cobedangiu   5
N 2 hours ago by Cobedangiu
Source: own
$a,b>0$ and $a+b=1$. Find min P:
$P=\sqrt{\frac{1-a}{1+7a}}+\sqrt{\frac{1-b}{1+7b}}$
5 replies
Cobedangiu
Yesterday at 6:10 PM
Cobedangiu
2 hours ago
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Proving ∠BHF=90
BarisKoyuncu   17
N 3 hours ago by jordiejoh
Source: IGO 2021 Advanced P1
Acute-angled triangle $ABC$ with circumcircle $\omega$ is given. Let $D$ be the midpoint of $AC$, $E$ be the foot of altitude from $A$ to $BC$, and $F$ be the intersection point of $AB$ and $DE$. Point $H$ lies on the arc $BC$ of $\omega$ (the one that does not contain $A$) such that $\angle BHE=\angle ABC$. Prove that $\angle BHF=90^\circ$.
17 replies
BarisKoyuncu
Dec 30, 2021
jordiejoh
3 hours ago
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Trapezium inscribed in a circle
shivangjindal   27
N 3 hours ago by andrewthenerd
Source: Balkan Mathematics Olympiad 2014 - Problem-3
Let $ABCD$ be a trapezium inscribed in a circle $\Gamma$ with diameter $AB$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$ . The circle with center $B$ and radius $BE$ meets $\Gamma$ at the points $K$ and $L$ (where $K$ is on the same side of $AB$ as $C$). The line perpendicular to $BD$ at $E$ intersects $CD$ at $M$. Prove that $KM$ is perpendicular to $DL$.

Greece - Silouanos Brazitikos
27 replies
shivangjindal
May 4, 2014
andrewthenerd
3 hours ago
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Tangent Spheres and Tangents to Spheres
Math-Problem-Solving   0
3 hours ago
Source: 2002 British Mathematical Olympiad Round 2
Prove this.
0 replies
1 viewing
Math-Problem-Solving
3 hours ago
0 replies
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INAMO 2019 P7
GorgonMathDota   7
N 3 hours ago by SYBARUPEMULA
Source: INAMO 2019 P7
Determine all solutions of
\[ x + y^2 = p^m \]\[ x^2 + y = p^n \]For $x,y,m,n$ positive integers and $p$ being a prime.
7 replies
GorgonMathDota
Jul 3, 2019
SYBARUPEMULA
3 hours ago
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Interesting inequalities
sqing   5
N 3 hours ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ a+b+c=3 $. Prove that
$$ \frac{a^2}{a^2+b+c+ \frac{3}{2}}+\frac{b^2}{b^2+c+a+\frac{3}{2}}+\frac{c^2}{c^2+a+b+\frac{3}{2}} \leq \frac{6}{7}$$Equality holds when $ (a,b,c)=(0,\frac{3}{2},\frac{3}{2}) $ or $ (a,b,c)=(0,0,3) .$
5 replies
sqing
5 hours ago
sqing
3 hours ago
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IGO 2022 advanced/free P1
Tafi_ak   11
N 3 hours ago by jordiejoh
Source: Iranian Geometry Olympiad 2022 P1 Advanced, Free
Four points $A$, $B$, $C$ and $D$ lie on a circle $\omega$ such that $AB=BC=CD$. The tangent line to $\omega$ at point $C$ intersects the tangent line to $\omega$ at $A$ and the line $AD$ at $K$ and $L$. The circle $\omega$ and the circumcircle of triangle $KLA$ intersect again at $M$. Prove that $MA=ML$.

Proposed by Mahdi Etesamifard
11 replies
Tafi_ak
Dec 13, 2022
jordiejoh
3 hours ago
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max q such that exists X in plane of equilateral ABC with AX: BX:CX=1:q:q^2
parmenides51   2
N Jun 27, 2021 by parmenides51
Source: 2017 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p10
Find the largest number $q$ for which there is a point $X$ in the plane of an equilateral triangle $ABC$ such that $AX: BX: CX = 1: q: q^2$.

(W. Bryman)
2 replies
parmenides51
Jun 27, 2021
parmenides51
Jun 27, 2021
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max q such that exists X in plane of equilateral ABC with AX: BX:CX=1:q:q^2
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Reveal topic tags
max
geometry
Equilateral
geometric inequality
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Source: 2017 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p10
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parmenides51
30629 posts
parmenides51
#1 h  Jun 27, 2021, 8:42 PM
Y by
Find the largest number $q$ for which there is a point $X$ in the plane of an equilateral triangle $ABC$ such that $AX: BX: CX = 1: q: q^2$.

(W. Bryman)
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cadaeibf
700 posts
cadaeibf
#2 h  Jun 27, 2021, 9:34 PM
Y by
Such a point would need to lie on two Apollonius circumferences, with centers on $AB$ and $BC$ respectively, which in the optimal case must be tangent to eachother. Since the ratios of this two Apollonius circumferences must both be $1:q$. The two centers $O$ and $O'$ must lie on the extensions of $BA$ and $CB$ such that $OA=OB=\frac{1}{q^2-1}$ and the diameters must both be $\frac{2q}{q^2-1}$.
Now, we can use cosine law on $OO'B$ to get $\left(\frac{q^2}{q^2-1}\right)^2+\left(\frac{1}{(q^2-1)}\right)^2+\frac{1(q^2)}{(q^2-1)^2}=\left(\frac{2q}{q^2-1}\right)^2\implies q^4-3q^2+1=0 \implies q=\sqrt{\frac{3\pm\sqrt{5}}{2}}=\frac{1\pm\sqrt{5}}{2}$ and so if $q>1$ the answer is $q=\frac{1+\sqrt{5}}{2}=\varphi$, i.e. the golden ratio.
This post has been edited 2 times. Last edited by cadaeibf, Jun 27, 2021, 10:20 PM
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parmenides51
30629 posts
parmenides51
#3 h  Jun 27, 2021, 9:40 PM
Y by
the result is
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