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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
43 minutes ago
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 16th (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
43 minutes ago
0 replies
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H
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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Assisted perpendicular chasing
sarjinius   3
N 2 minutes ago by ZeroHero
Source: Philippine Mathematical Olympiad 2025 P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.
3 replies
sarjinius
Mar 9, 2025
ZeroHero
2 minutes ago
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(a²-b²)(b²-c²) = abc
straight   4
N 8 minutes ago by GreekIdiot
Find all triples of positive integers $(a,b,c)$ such that

\[(a^2-b^2)(b^2-c^2) = abc.\]
If you can't solve this, assume $gcd(a,c) = 1$. If this is still too hard assume in $a \ge b \ge c$ that $b-c$ is a prime.
4 replies
straight
Mar 24, 2025
GreekIdiot
8 minutes ago
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Prove that there are no tuples $(x, y, z)$ sastifying $x^2+y^2-z^2=xyz-2$
Anabcde   2
N 12 minutes ago by IMUKAT
Prove that there are no tuples $(x, y, z) \in \mathbb{Z}^3$ sastifying $x^2+y^2-z^2=xyz-2$
2 replies
Anabcde
3 hours ago
IMUKAT
12 minutes ago
J
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Old problem :(
Drakkur   0
23 minutes ago
Let a, b, c be positive real numbers. Prove that
$$\dfrac{1}{\sqrt{a^2+bc}}+\dfrac{1}{\sqrt{b^2+ca}}+\dfrac{1}{\sqrt{c^2+ab}}\le \sqrt{2}\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}\right)$$
0 replies
Drakkur
23 minutes ago
0 replies
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Tangent circles and equal angles
MRF2017   1
N Mar 21, 2016 by Kenny O
Source: Kazakhstan national olympiad 2016 Final Round,grade 11,P3
Circles $\omega_1 , \omega_2$ intersect at points $X,Y$ and they are internally tangent to circle $\Omega$ at points $A,B$,respectively.$AB$ intersect with $\omega_1 , \omega_2$ at points $A_1,B_1$ ,respectively.Another circle is internally tangent to $\omega_1 , \omega_2$ and $A_1B_1$ at $Z$.Prove that $\angle AXZ =\angle BXZ$.(C.Ilyasov)
1 reply
MRF2017
Mar 21, 2016
Kenny O
Mar 21, 2016
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Tangent circles and equal angles
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Reveal topic tags
geometry proposed
geometry
circles
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Source: Kazakhstan national olympiad 2016 Final Round,grade 11,P3
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MRF2017
237 posts
MRF2017
#1 h  Mar 21, 2016, 7:41 PM • 2 Y
Y by Adventure10, Mango247
Circles $\omega_1 , \omega_2$ intersect at points $X,Y$ and they are internally tangent to circle $\Omega$ at points $A,B$,respectively.$AB$ intersect with $\omega_1 , \omega_2$ at points $A_1,B_1$ ,respectively.Another circle is internally tangent to $\omega_1 , \omega_2$ and $A_1B_1$ at $Z$.Prove that $\angle AXZ =\angle BXZ$.(C.Ilyasov)
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Kenny O
116 posts
Kenny O
#2 h  Mar 21, 2016, 7:59 PM • 2 Y
Y by Adventure10, Mango247
Nice and easy.
Let H be the external similitude centrer of $w_1$ and $w_2$. By Monge 's theorem it lies on $AB$ so the inversion with pole H interchanges these two circles and fixes the small circle internally tangent to them so $HX^2=HZ^2=HA*HB$ so the circle with centre H and radius HY is Apollonian circle of $XAB$ and we are done. :)
This post has been edited 2 times. Last edited by Kenny O, Mar 21, 2016, 8:01 PM
Reason: Typo
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