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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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geometry
Tony_stark0094   0
4 minutes ago
Consider $\Delta ABC$ let $\omega_1$ and $\omega_2$ be the circles passing through $A,B$ and $A,C$ respectively such that $BC$ is tangent to $\omega_1$ and $\omega_2$ define $R$ to be a point such that it lies on both the circles $\omega_1$ and $\omega_2$ prove that $HR$ and $AR$ are perpendicular.
0 replies
Tony_stark0094
4 minutes ago
0 replies
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Inequalities
sqing   1
N an hour ago by DAVROS
Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=11.$ Prove that
$$a+ab+abc\leq\frac{49}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=10.$ Prove that
$$a+ab+abc\leq\frac{169}{24}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=14.$ Prove that
$$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=32.$ Prove that
$$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$$
1 reply
sqing
Yesterday at 2:59 PM
DAVROS
an hour ago
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Proving ∠BHF=90
BarisKoyuncu   17
N 2 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
2 hours ago
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Trapezium inscribed in a circle
shivangjindal   27
N 2 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
2 hours ago
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one very nice!
MihaiT   1
N 2 hours ago by MihaiT
Given $m_1$ weights, each weighing $k_1$ and another $m_2$ weights with $k_2$ each. Write a algorithm that determines the ways in which a scale can be balanced with a weight $X$ on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers $m_1,k_1,m_2,k_2,X$. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?

One answer is possible:
a)10;5;0;
b)20;7;0;
c)20;7;1;
d)10;10;0;
e)10;7;0;
f)20;5;0,
1 reply
MihaiT
Mar 31, 2025
MihaiT
2 hours ago
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Tangent Spheres and Tangents to Spheres
Math-Problem-Solving   0
2 hours ago
Source: 2002 British Mathematical Olympiad Round 2
Prove this.
0 replies
Math-Problem-Solving
2 hours ago
0 replies
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INAMO 2019 P7
GorgonMathDota   7
N 2 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
2 hours ago
J
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Interesting inequalities
sqing   5
N 2 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
4 hours ago
sqing
2 hours ago
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IGO 2022 advanced/free P1
Tafi_ak   11
N 2 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
2 hours ago
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Geo Mock #4
Bluesoul   1
N 2 hours ago by Sedro
Consider acute triangle $ABC$ with orthocenter $H$. Extend $AH$ to meet $BC$ at $D$. The angle bisector of $\angle{ABH}$ meets the midpoint of $AD$. If $AB=10, BH=6$, compute the area of $\triangle{ABC}$.
1 reply
Bluesoul
Yesterday at 7:03 AM
Sedro
2 hours ago
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Interesting inequalities
sqing   1
N 3 hours ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ a+b+c=3 $. Prove that
$$ \frac{a^2+1}{a^2+b+c-\frac{1}{2}}+\frac{b^2+1}{b^2+c+a-\frac{1}{2}}+\frac{c^2+1}{c^2+a+b-\frac{1}{2}} \leq \frac{12}{5}$$$$ \frac{a^2+2}{a^2+b+c+\frac{1}{2}}+\frac{b^2+2}{b^2+c+a+\frac{1}{2}}+\frac{c^2+2}{c^2+a+b+\frac{1}{2}} \leq \frac{18}{7}$$Equality holds when $ (a,b,c)=(0,\frac{3}{2},\frac{3}{2}) $ or $ (a,b,c)=(1,1,1) .$
1 reply
sqing
3 hours ago
sqing
3 hours ago
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euler line and midpoint tangents
Thelink_20   3
N 3 hours ago by hectorleo123
Source: My problem
Let the medians of a triangle $\Delta ABC$ intersect its circumcircle $\Gamma$ at $N_A, N_B, N_C$. The tangets to $\Gamma$ from $N_A,N_B,N_C$ determine a triangle $\Delta X_AX_BX_C$, where $X_A$ is relative to $A$ and so on. Prove that lines $AX_A,BX_B,CX_C$ are concurrent at a point $P$ and that $P$ belongs to the euler line of $\Delta ABC$.
Remark:Click to reveal hidden text

3 replies
Thelink_20
Jan 24, 2025
hectorleo123
3 hours ago
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Problem 4: ISL 2008/G3 Constructed Four Times
ike.chen   25
N 4 hours ago by Yiyj1
Source: USEMO 2022/4
Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points $P, Q, R, S$ lie in the interiors of segments $AB, BC, CD, DA,$ respectively, such that $$\angle PDA = \angle PCB, \text{ } \angle QAB = \angle QDC, \text{ } \angle RBC = \angle RAD, \text{ and } \angle SCD = \angle SBA.$$Let $AQ$ intersect $BS$ at $X$, and $DQ$ intersect $CS$ at $Y$. Prove that lines $PR$ and $XY$ are either parallel or coincide.

Tilek Askerbekov
25 replies
ike.chen
Oct 23, 2022
Yiyj1
4 hours ago
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Inspired by old results
sqing   8
N 4 hours ago by sqing
Source: Own
Let $ a,b,c> 0 $ and $ abc=1 $. Prove that
$$\frac1{a^2+a+k}+\frac1{b^2+b+k}+\frac1{c^2+c+k}\geq \frac{3}{k+2}$$Where $ 0<k \leq 1.$
8 replies
sqing
Monday at 1:42 PM
sqing
4 hours ago
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J
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2017 preRMO p27, circle lies on interior of another circle, radius wanted
parmenides51   3
N Jul 1, 2021 by Commander_Anta78
Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and B1 respectively such that $A_1$ and $B_1$ are on the opposite sides of $AB$. Given that $A_1 B = 5, AB_1 = 15$ and $OP = 10$, find the radius of $\Omega_1$.
3 replies
parmenides51
Aug 20, 2019
Commander_Anta78
Jul 1, 2021
J
2017 preRMO p27, circle lies on interior of another circle, radius wanted
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Reveal topic tags
geometry
circle
radius
circles
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parmenides51
30629 posts
parmenides51
#1 h  Aug 20, 2019, 1:09 PM • 1 Y
Y by Adventure10
Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and B1 respectively such that $A_1$ and $B_1$ are on the opposite sides of $AB$. Given that $A_1 B = 5, AB_1 = 15$ and $OP = 10$, find the radius of $\Omega_1$.
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Commander_Anta78
58 posts
Commander_Anta78
#2 h  Oct 5, 2020, 1:43 PM • 1 Y
Y by Mango247
seems interesting
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Ansh1908
8 posts
Ansh1908
#3 h  Oct 6, 2020, 8:52 PM • 2 Y
Y by Mango247, Mango247
Let $AA_1$ and $BB_1$ be tangents to $\Omega_2$ at $A_2$ and $B_2$ respectively.
Now note that $\angle AA_2P=\angle AA_1B=\angle BB_2P=\angle BB_1A= \frac{\pi}{2}$

We now have $\frac{AO+OP}{PA_2}=\frac{AB}{A_1B}$
and that $\frac{OB-OP}{PB_2}=\frac{AB}{AB_1}$
Combining these two and substituting the given values, we get $OA=20$.
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Commander_Anta78
58 posts
Commander_Anta78
#4 h  Jul 1, 2021, 2:52 PM • 1 Y
Y by Mango247
/Bump

Also 1st post after a month
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