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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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Reflections of AB, AC with respect to BC and angle bisector of A
falantrng   26
N 4 minutes ago by ehuseyinyigit
Source: BMO 2024 Problem 1
Let $ABC$ be an acute-angled triangle with $AC > AB$ and let $D$ be the foot of the
$A$-angle bisector on $BC$. The reflections of lines $AB$ and $AC$ in line $BC$ meet $AC$ and $AB$ at points
$E$ and $F$ respectively. A line through $D$ meets $AC$ and $AB$ at $G$ and $H$ respectively such that $G$
lies strictly between $A$ and $C$ while $H$ lies strictly between $B$ and $F$. Prove that the circumcircles of
$\triangle EDG$ and $\triangle FDH$ are tangent to each other.
26 replies
falantrng
Apr 29, 2024
ehuseyinyigit
4 minutes ago
J
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configurational geometry as usual
GorgonMathDota   11
N 12 minutes ago by ratavir
Source: Indonesia National Math Olympiad 2021 Problem 7 (INAMO 2021/7)
Given $\triangle ABC$ with circumcircle $\ell$. Point $M$ in $\triangle ABC$ such that $AM$ is the angle bisector of $\angle BAC$. Circle with center $M$ and radius $MB$ intersects $\ell$ and $BC$ at $D$ and $E$ respectively, $(B \not= D, B \not= E)$. Let $P$ be the midpoint of arc $BC$ in $\ell$ that didn't have $A$. Prove that $AP$ angle bisector of $\angle DPE$ if and only if $\angle B = 90^{\circ}$.
11 replies
GorgonMathDota
Nov 9, 2021
ratavir
12 minutes ago
J
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kind of well known?
dotscom26   1
N 44 minutes ago by dotscom26
Source: MBL
Let $ y_1, y_2, ..., y_{2025}$ be real numbers satisfying
$ y_1^2 + y_2^2 + \cdots + y_{2025}^2 = 1. $
Find the maximum value of
$ |y_1 - y_2| + |y_2 - y_3| + \cdots + |y_{2025} - y_1|. $

I have seen many problems with the same structure, Id really appreciate if someone could explain which approach is suitable here
1 reply
+1 w
dotscom26
Today at 4:11 AM
dotscom26
44 minutes ago
J
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sum(ab/4a^2+b^2) <= 3/5
truongphatt2668   2
N an hour ago by arqady
Source: I remember I read it somewhere
Let $a,b,c>0$. Prove that:
$$\dfrac{ab}{a^2+4b^2} + \dfrac{bc}{b^2+4c^2} + \dfrac{ca}{c^2+4a^2} \le \dfrac{3}{5}$$
2 replies
1 viewing
truongphatt2668
Yesterday at 1:23 PM
arqady
an hour ago
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No more topics!
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J
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nine-point circle generalized
jgnr   10
N Oct 14, 2024 by Eka01
Source: bulgaria 2011
Point $O$ is inside $\triangle ABC$. The feet of perpendicular from $O$ to $BC,CA,AB$ are $D,E,F$. Perpendiculars from $A$ and $B$ respectively to $EF$ and $FD$ meet at $P$. Let $H$ be the foot of perpendicular from $P$ to $AB$. Prove that $D,E,F,H$ are concyclic.
10 replies
jgnr
May 30, 2011
Eka01
Oct 14, 2024
J
nine-point circle generalized
G H J
Reveal topic tags
geometry
circumcircle
geometry proposed
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G H BBookmark kLocked kLocked NReply
Source: bulgaria 2011
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jgnr
1343 posts
jgnr
#1 h  May 30, 2011, 3:17 AM • 4 Y
Y by anhtaitran, Adventure10, Mango247, ehuseyinyigit
Point $O$ is inside $\triangle ABC$. The feet of perpendicular from $O$ to $BC,CA,AB$ are $D,E,F$. Perpendiculars from $A$ and $B$ respectively to $EF$ and $FD$ meet at $P$. Let $H$ be the foot of perpendicular from $P$ to $AB$. Prove that $D,E,F,H$ are concyclic.
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jayme
9774 posts
jayme
#2 h  May 30, 2011, 5:31 AM • 2 Y
Y by AlastorMoody, Adventure10
Dear Mathlinkers,
this nice result can be solved in this way
1. the pedal circle theorem
2. the isogonal theorem
Sincerely
Jean-Louis
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Vo Duc Dien
341 posts
Vo Duc Dien
#3 h  May 30, 2011, 5:43 AM • 2 Y
Y by Adventure10, Mango247
You have the rest of the Bulgaria MO 2011 problem?
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jgnr
1343 posts
jgnr
#4 h  May 30, 2011, 2:02 PM • 1 Y
Y by Adventure10
You're right jayme. Moreover, we can prove that the circumcenter of $DEF$ is the midpoint of $OP$.

I found Bulgaria 2011 problems here.
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Vo Duc Dien
341 posts
Vo Duc Dien
#5 h  Oct 31, 2011, 4:24 AM • 1 Y
Y by Adventure10
JPNE is cyclic (PN⊥BC, N on BC) and the problem is solved.

This modified problem is harder:

Let K, Π, L and M be the intersection of OE and HP, the circumcircle of triangle DEH, the circumcenter of Π, and the circumcenter of Γ. Prove that ∠OBK = ∠DML.
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Sayan
2130 posts
Sayan
#6 h  Jan 29, 2012, 4:15 AM • 2 Y
Y by Adventure10, Mango247
Johan Gunardi wrote:
The feet of perpendicular from $O$ to $AB,BC,CA$ are $D,E,F$.

i think it should be $BC,CA,AB$ are $D,E,F$
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nima-amini
184 posts
nima-amini
#7 h  Dec 31, 2013, 1:48 PM • 2 Y
Y by Adventure10, Mango247
use six point circle
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Mathematicalx
537 posts
Mathematicalx
#8 h  Jul 8, 2014, 2:27 PM • 3 Y
Y by NOLF, Adventure10, Mango247
Can someone give solution?
All things above doesnt make sense.
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SidVicious
584 posts
SidVicious
#9 h  Sep 11, 2016, 8:24 PM • 2 Y
Y by Adventure10, Mango247
Let $P', P''$ be feet of perpendiculars from $P$ to $BC,CA$, respectively.
We have $\angle PAE=\angle FEO=\angle OAF$ and similarly $\angle OBA=\angle PBC$ $\implies$ $O,P$ are isogonal conjugates of $\Delta ABC$.
It is well known that pedal triangles of isogonal conjugates share the circumcircle $\implies$ $HFDP'P''E$ is cyclic $\implies$ $EHFD$ is cyclic.
This post has been edited 1 time. Last edited by SidVicious, Sep 11, 2016, 8:24 PM
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jbaca
225 posts
jbaca
#10 h  Apr 26, 2020, 11:34 PM
Y by
Solution. Since $AO$ is a diameter of $(AEF)$ and $AP$ is the $A$-altitude of $\bigtriangleup AEF$, we conclude that $AO$ and $AP$ are isogonal conjugates of $\bigtriangleup AEF$ and $\bigtriangleup ABC$. We obtain similar results for $BO,\ BP$ and $CO,\ CP$, then $O$ and $P$ are isogonal conjugates of $\bigtriangleup ABC$ and the result follows from the pedal circle lemma. $\blacksquare$
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Eka01
204 posts
Eka01
#11 h  Oct 14, 2024, 2:21 PM
Y by
It is well known that $P$ is the isogonal conjugate of $O$ and it is also known that feet of perpendiculars to the sides lie on the same circle and hence we are done.
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