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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
1 viewing
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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sum(ab/4a^2+b^2) <= 3/5
truongphatt2668   3
N 5 minutes 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
1 viewing
truongphatt2668
Monday at 1:23 PM
Nguyenhuyen_AG
5 minutes ago
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Inspired by old results
sqing   2
N 15 minutes ago by sqing
Source: Own
Let $ a,b $ be reals such that $ a^2+b^2 +ab =1.$ Prove that $$9\geq (a^2-a+1)(b^2-b+1) (a^2-ab+b^2) \geq \frac{19-8\sqrt 3}{27}$$Let $ a,b $ be reals such that $ a^3+b^3 -2ab =1.$ Prove that $$ (a^2-a+1)(b^2-b+1) (a^2-ab+b^2) \geq 1$$
2 replies
1 viewing
sqing
23 minutes ago
sqing
15 minutes ago
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Painting Beads on Necklace
amuthup   45
N an hour ago by maromex
Source: 2021 ISL C2
Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible.

Carl Schildkraut, USA
45 replies
amuthup
Jul 12, 2022
maromex
an hour ago
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Inclusion Exclusion Principle
chandru1   1
N an hour ago by onofre.campos
How does one prove the identity $$1=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}2^{n-k}$$This easy via the binomial theorem for the quantity is just $(2-1)^{k}$, but how do we arrive at this using the I-E-P?
1 reply
chandru1
Dec 4, 2020
onofre.campos
an hour ago
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No more topics!
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Proving the equilateral triangle
CinarArslan   1
N Dec 23, 2019 by NahTan123xyz
Source: Turkey National Mathematical Olympiad 2019 P4
In a triangle $\Delta ABC$, $|AB|=|AC|$. Let $M$ be on the minor arc $AC$ of the circumcircle of $\Delta ABC$ different than $A$ and $C$. Let $BM$ and $AC$ meet at $E$ and the bisector of $\angle BMC$ and $BC$ meet at $F$ such that $\angle AFB=\angle CFE$. Prove that the triangle $\Delta ABC$ is equilateral.
1 reply
CinarArslan
Dec 23, 2019
NahTan123xyz
Dec 23, 2019
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Proving the equilateral triangle
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Reveal topic tags
geometry
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G H BBookmark kLocked kLocked NReply
Source: Turkey National Mathematical Olympiad 2019 P4
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CinarArslan
143 posts
CinarArslan
#1 h  Dec 23, 2019, 4:57 AM • 3 Y
Y by Adventure10, Mango247, Creeper1612
In a triangle $\Delta ABC$, $|AB|=|AC|$. Let $M$ be on the minor arc $AC$ of the circumcircle of $\Delta ABC$ different than $A$ and $C$. Let $BM$ and $AC$ meet at $E$ and the bisector of $\angle BMC$ and $BC$ meet at $F$ such that $\angle AFB=\angle CFE$. Prove that the triangle $\Delta ABC$ is equilateral.
This post has been edited 2 times. Last edited by CinarArslan, Dec 23, 2019, 4:58 AM
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NahTan123xyz
372 posts
NahTan123xyz
#2 h  Dec 23, 2019, 7:59 AM • 3 Y
Y by raven_, Adventure10, Creeper1612
CinarArslan wrote:
In a triangle $\Delta ABC$, $|AB|=|AC|$. Let $M$ be on the minor arc $AC$ of the circumcircle of $\Delta ABC$ different than $A$ and $C$. Let $BM$ and $AC$ meet at $E$ and the bisector of $\angle BMC$ and $BC$ meet at $F$ such that $\angle AFB=\angle CFE$. Prove that the triangle $\Delta ABC$ is equilateral.

It's easy to see that $\triangle ABF \sim+ \triangle ECF$.
So we have $$\frac{AB}{EC}=\frac{BF}{CF}=\frac{BM}{CM}.$$But $\angle ABM=\angle ECM$. Therefore, $\triangle ABM \sim+ \triangle ECM$.
Hence, $\angle ACB=\angle AMB=\angle EMC=\angle BAC$. So $\triangle ABC$ must be equilateral.
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