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a My Retirement & New Leadership at AoPS
rrusczyk   1401
N a few seconds ago by Solocraftsolo
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
1401 replies
rrusczyk
Monday at 6:37 PM
Solocraftsolo
a few seconds ago
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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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Inequalities
sqing   6
N 22 minutes ago by DAVROS
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +21abc\leq\frac{512}{441}$$Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$$Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$$Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$
6 replies
sqing
Today at 3:07 AM
DAVROS
22 minutes ago
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Girls in Math at Yale 2025 Team P12:tangent circles
Bluesoul   3
N 32 minutes ago by ehuseyinyigit
Consider triangle $ABC$ with $AB=5, BC=7, AC=8$. Denote the incircle of $\triangle{ABC}$ as $\omega$ and let $\omega$ meet $AB,AC,BC$ at $D,E,F$ respectively. Draw circle $\gamma$ such that $\gamma$ passes through $A$ and tangents to $BC$ at $F$. Denote the intersections of line $DE$ and $\gamma$ as $X$ and $Y$, the length of $XY$ can be written in the simplest form of $\frac{p\sqrt{q}}{r}$, compute $p+q+r$.
3 replies
Bluesoul
Today at 8:49 AM
ehuseyinyigit
32 minutes ago
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Find the midpoint of the chord of a conic
Hunter87   2
N an hour ago by Hunter87
From P(4,5), the chord of contact to the conic 3x² + 4y² = 1 is AB, we are to find the midpoint of this chord.

I used T(4,5)=0 to get eqn. of AB, then assuming (h,k) to be the midpoint, T(h,k)=S1(h,k) should give the equation of AB again. But comparing both equations to get h,k does not give me the correct answer.

What am I doing wrong?
2 replies
Hunter87
Today at 8:15 AM
Hunter87
an hour ago
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3D Geometry Problem
ReticulatedPython   1
N an hour ago by ReticulatedPython
Three mutually tangent non-degenerate spheres rest on a plane. Let their centers be $A, B$, and $C$. The spheres with centers $A, B$, and $C$ touch the plane at $P, Q$, and $R$, respectively. Prove that $$\frac{1}{AP}+\frac{1}{BQ}+\frac{1}{CR}+PQ+RQ+PR \ge 6\sqrt{2}$$
1 reply
ReticulatedPython
Yesterday at 8:12 PM
ReticulatedPython
an hour ago
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No more topics!
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Multivariable Factor Theorem
chocopuff   8
N Apr 17, 2016 by 62861
Is there a quick way to factor this polynomial? $x^3+y^3+z^3-3xyz$ just by noticing that if $x=y=z$, the polynomial equals $0$.
8 replies
chocopuff
Feb 6, 2016
62861
Apr 17, 2016
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Multivariable Factor Theorem
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Reveal topic tags
algebra
polynomial
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chocopuff
401 posts
chocopuff
#1 h  Feb 6, 2016, 1:08 PM • 2 Y
Y by Adventure10, Mango247
Is there a quick way to factor this polynomial? $x^3+y^3+z^3-3xyz$ just by noticing that if $x=y=z$, the polynomial equals $0$.
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62861
3564 posts
62861
#2 h  Feb 6, 2016, 1:12 PM • 2 Y
Y by Adventure10, Mango247
Not just by noticing that. However, you can factor it by noticing that the polynomial equals $0$ if $x+y+z=0$.

In fact, the polynomial
\[a_1^n+a_2^n+\cdots+a_n^n-na_1a_2\cdots a_n\]is irreducible for all $n\geq4$.
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jlammy
1099 posts
jlammy
#3 h  Feb 6, 2016, 1:31 PM • 2 Y
Y by Adventure10, Mango247
No, but it may help to notice that $x=-y-z$ is a root, so the equation has a factor of $x+y+z$. See this thread for a discussion on the factorisation (starting from post 10).
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shmm
626 posts
shmm
#4 h  Feb 6, 2016, 9:00 PM • 1 Y
Y by Adventure10
also : $x^3+y^3+z^3-3xyz=(x+y+z)(x^2 + y^2 + z^2-xy-yz-zx)$
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ThE-dArK-lOrD
4071 posts
ThE-dArK-lOrD
#5 h  Feb 7, 2016, 6:15 AM • 2 Y
Y by Adventure10, Mango247
Also $x^3+y^3+z^3-3xyz=\frac{1}{2}(x+y+z)((x-y)^2+(y-z)^2+(z-x)^2)$
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untimate
28 posts
untimate
#6 h  Apr 13, 2016, 2:09 PM • 2 Y
Y by Adventure10, Mango247
$x^3+y^3+z^3-3xyz=(x+y+z)(x^2 + y^2 + z^2-xy-yz-zx)$
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Virgil Nicula
7054 posts
Virgil Nicula
#7 h  Apr 16, 2016, 10:13 PM • 2 Y
Y by Adventure10, Mango247
See PP13 from here
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Not_a_Username
1215 posts
Not_a_Username
#8 h  Apr 17, 2016, 4:04 PM • 2 Y
Y by Adventure10, Mango247
Virgil Nicula wrote:
See PP13 from here

??? Leads to this which has nothing to do with the topic at hand.

EDIT: disregard this, misunderstanding
This post has been edited 1 time. Last edited by Not_a_Username, Apr 17, 2016, 5:21 PM
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62861
3564 posts
62861
#9 h  Apr 17, 2016, 4:13 PM • 2 Y
Y by Adventure10, Mango247
Not_a_Username wrote:
Virgil Nicula wrote:
See PP13 from here

??? Leads to this which has nothing to do with the topic at hand.

Scroll down on his post 440.
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