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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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prove that $\angle Q L A=\angle M L A$
NJAX   3
N a minute ago by Baimukh
Source: 2nd Al-Khwarizmi International Junior Mathematical Olympiad 2024, Day2, Problem7
Two circles with centers $O_{1}$ and $O_{2}$ intersect at $P$ and $Q$. Let $\omega$ be the circumcircle of the triangle $P O_{1} O_{2}$; the circle $\omega$ intersect the circles centered at $O_{1}$ and $O_{2}$ at points $A$ and $B$, respectively. The point $Q$ is inside triangle $P A B$ and $P Q$ intersects $\omega$ at $M$. The point $E$ on $\omega$ is such that $P Q=Q E$. Let $M E$ and $A B$ meet at $L$, prove that $\angle Q L A=\angle M L A$.

Proposed by Amir Parsa Hoseini Nayeri, Iran
3 replies
NJAX
May 31, 2024
Baimukh
a minute ago
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Three Nagel points collinear
jayme   3
N 4 minutes ago by buratinogigle
Dear Marthlinkers,

1. ABCD a square
2. M a point on the segment CD sothat MA < MB
3. Nm, Na, Nb the Nagel’s points of the triangles MAD, ADM, BCM.

Prove : Nm, Na and Nb are collinear.

Sincerely
Jean-Louis
3 replies
jayme
Mar 31, 2025
buratinogigle
4 minutes ago
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How many acute triangles can a right isosceles triangle be divided into?
truongphatt2668   3
N 8 minutes ago by truongphatt2668
Source: I don't know
How many acute triangles can a right isosceles triangle be divided into?
3 replies
truongphatt2668
Yesterday at 2:48 PM
truongphatt2668
8 minutes ago
J
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sum(ab/4a^2+b^2) <= 3/5
truongphatt2668   4
N 10 minutes ago by truongphatt2668
Source: I remember I read it somewhere
Let $a,b,c>0$. Prove that:
$$\dfrac{ab}{a^2+4b^2} + \dfrac{bc}{b^2+4c^2} + \dfrac{ca}{c^2+4a^2} \le \dfrac{3}{5}$$
4 replies
truongphatt2668
Monday at 1:23 PM
truongphatt2668
10 minutes ago
J
No more topics!
H
J
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Show that KM is bisector
SymbatK   5
N Jan 31, 2015 by Sardor
Source: Kazakhstan NMO 2009 9th grade
Given triangle ABC,where AB>AC.Altitudes CC1 and BB1 intersect at H,lines B1C1 and BC at P.Let M be midpoint of BC,line segments MH & AP intersect each other at K respectively.Prove that KM is angle bisector of B1KB.
5 replies
SymbatK
Jan 31, 2015
Sardor
Jan 31, 2015
J
Show that KM is bisector
G H J
Reveal topic tags
geometry
angle bisector
geometry unsolved
L
G H BBookmark kLocked kLocked NReply
Source: Kazakhstan NMO 2009 9th grade
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SymbatK
13 posts
SymbatK
#1 h  Jan 31, 2015, 1:53 PM • 1 Y
Y by Adventure10
Given triangle ABC,where AB>AC.Altitudes CC1 and BB1 intersect at H,lines B1C1 and BC at P.Let M be midpoint of BC,line segments MH & AP intersect each other at K respectively.Prove that KM is angle bisector of B1KB.
Z K Y
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jayme
9775 posts
jayme
#2 h  Jan 31, 2015, 2:09 PM • 1 Y
Y by Adventure10
Dear Mathlinkers,
1. it is known that MK is perpendicular to AP
2. We have to prove that we have an harmonic pencil...

Sincerely
Jean-Louis
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Omez
55 posts
Omez
#3 h  Jan 31, 2015, 2:12 PM • 1 Y
Y by Adventure10
See that AP is the polar of H in the circle on diameter BC. So $MH \bot AP$. Name $D$ the intersection of $BB_1$ and $AP$. From the harmonic bundle $ (BB_1,HD)=-1$ we get KM is angle bisector of <B1KB so done.
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ATimo
228 posts
ATimo
#4 h  Jan 31, 2015, 2:25 PM • 4 Y
Y by sasanineq, Adventure10, Mango247, and 1 other user
My solution:
$B1C1BC$ is concyclic.So $K$ is the Miquel point of $B1C1BC$ and we have $HKA=90$. So $AKB1HC1$ is concyclic. So $HKB1=HAB1=B1BM$. So $B1KBM$ is concyclic. $B1M=BM$, So we have $B1KM=MKB$.
And we are done.
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SymbatK
13 posts
SymbatK
#5 h  Jan 31, 2015, 3:00 PM • 2 Y
Y by Adventure10, Mango247
Has anyone solution without poles and polars?
thanks)
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Sardor
804 posts
Sardor
#6 h  Jan 31, 2015, 4:49 PM • 2 Y
Y by Adventure10, Mango247
We know that $ H $ is orthocenter of $ APM $ and $ AKB_1HC_1 $ is cyclic.It's easy to see that we must prove that $ \angle BKH=\angle B_1BM $ or $ MB^2=MH \cdot MK $, but it's exactly true !
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