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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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IMO 2008, Question 1
orl   153
N 4 minutes 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
4 minutes ago
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P(x) doesn't have 2024 distinct real roots
gatnghiep   1
N 6 minutes 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
gatnghiep
Oct 2, 2024
kiyoras_2001
6 minutes ago
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A lies on the radical axis of BQX and CPX
a_507_bc   35
N 35 minutes 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
35 minutes ago
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sum(ab/4a^2+b^2) <= 3/5
truongphatt2668   3
N an hour 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
an hour ago
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No more topics!
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Concyclicity
utkarshgupta   4
N Jul 6, 2014 by jlammy
Source: BrMO 2010/11 R2
Let $ABC$ be a triangle and $X$ be a point inside the triangle. The lines $AX,BX$ and $CX$ meet the circle $ABC$ again at $ P,Q$ and $R$ respectively. Choose a point $U$ on $XP$ which is between $X$ and $P$. Suppose that the lines through $U$ which are parallel to $AB$ and $CA$ meet $XQ$ and $XR$ at points $V$ and $W$ respectively. Prove that the points $R,W, V$ and $Q$ lie on a circle.
4 replies
utkarshgupta
Jul 6, 2014
jlammy
Jul 6, 2014
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Concyclicity
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Reveal topic tags
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geometry proposed
geometry
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Source: BrMO 2010/11 R2
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utkarshgupta
2280 posts
utkarshgupta
#1 h  Jul 6, 2014, 7:28 AM • 1 Y
Y by Adventure10
Let $ABC$ be a triangle and $X$ be a point inside the triangle. The lines $AX,BX$ and $CX$ meet the circle $ABC$ again at $ P,Q$ and $R$ respectively. Choose a point $U$ on $XP$ which is between $X$ and $P$. Suppose that the lines through $U$ which are parallel to $AB$ and $CA$ meet $XQ$ and $XR$ at points $V$ and $W$ respectively. Prove that the points $R,W, V$ and $Q$ lie on a circle.
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hajimbrak
209 posts
hajimbrak
#2 h  Jul 6, 2014, 8:29 AM • 2 Y
Y by Adventure10, Mango247
We have triangles $XWU$ and $XAC$ similar.Hence $\frac{XU}{XA}=\frac{XW}{XC}$.Similarly $\frac{XU}{XA}=\frac{XV}{XB}$ implying
$\frac{XW}{XV}=\frac{XC}{XB}$.Also $XQ.XB=XC.XR$.Combining the last two relations we get $XW.XR=XV.XQ$ implying the concyclicity of $R,W,V,Q$.
NOTE:It must be specified that $V$ and $W$ lie on the segments $XQ$ and $XR$ respectively.
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jlammy
1099 posts
jlammy
#3 h  Jul 6, 2014, 4:45 PM • 2 Y
Y by Adventure10, Mango247
Applying Desargues' theorem on $\triangle{ABC}$ and $\triangle{UVW}$ implies that the perspector line is at infinity, so that $VW \cap BC$ is also at infinity: in other words, $VW \parallel BC$.

$\therefore \angle{QRC} \&= \angle{QBC} = \angle{BVW}$

$\therefore$ $R, W, V$ and $Q$ lie on a circle, as required.

See http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=388732&p=2159730.

@utkarshgupta

This is not the first time you have created an already-existent topic on a BrMO question. The BrMO is quite a popular paper (I participate in it), so it would be wise to search the problem up to see if there is a topic on the question. If there is, which should normally be the case, then it is not wise to create such a topic.

Posting such old and popular questions is generally not wise, considering that in the (large) time space between the questions being released and your posting time, many such topics may have already been created, each with much discussion. For example, I know that AndrewTom usually posts BrMO questions on AoPS within 10 minutes of them being officially released on the BrMO website; i.e., within 10 minutes of internet discussion on the questions being permitted.
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utkarshgupta
2280 posts
utkarshgupta
#4 h  Jul 6, 2014, 5:07 PM • 2 Y
Y by Adventure10, Mango247
Ok Thanx!
I will try to perform a search a next time.

Thanx!
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jlammy
1099 posts
jlammy
#5 h  Jul 6, 2014, 6:48 PM • 2 Y
Y by Adventure10, Mango247
Yeah, you've reposted loads. Try searching AoPS for previous topics on:

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=596641&p=3540476#p3540476
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=596539&p=3540009#p3540009
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=595684&p=3539638#p3539638
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=592769
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&t=592770
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=592014&p=3539059#p3539059
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=590104&p=3538235#p3538235
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=592769&p=3538197#p3538197
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&t=592770&p=3538195#p3538195
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=592768&p=3528255#p3528255

and basically every other old BrMO question that you have posted: those are just the ones that I have responded to.
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