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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
2 hours 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
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[*]April 16th (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
2 hours ago
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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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Number theory
Maaaaaaath   0
a minute ago
Let $m$ be a positive integer . Prove that there exists infinitely many pairs of positive integers $(x,y)$ such that $\gcd(x,y)=1$ and :

$$xy | x^2+y^2+m$$
0 replies
Maaaaaaath
a minute ago
0 replies
J
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Only consecutive terms are coprime
socrates   35
N 18 minutes ago by zuat.e
Source: 7th RMM 2015, Problem 1
Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?
35 replies
socrates
Feb 28, 2015
zuat.e
18 minutes ago
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D1019 : Dominoes 2*1
Dattier   4
N 21 minutes ago by Dattier
I have a 9*9 grid like this one:

IMAGE

We choose 5 white squares on the lower triangle, 5 black squares on the upper triangle and one on the diagonal, which we remove from the grid.
Like for example here:

IMAGE

Can we completely cover the grid remove from these 11 squares with 2*1 dominoes like this one:

IMAGE
4 replies
Dattier
Mar 26, 2025
Dattier
21 minutes ago
J
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inequalities hard
Cobedangiu   4
N 24 minutes ago by Cobedangiu
problem
4 replies
Cobedangiu
Mar 31, 2025
Cobedangiu
24 minutes ago
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No more topics!
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2008 JBMO Shortlist G8
parmenides51   4
N May 6, 2023 by StarLex1
Source: 2008 JBMO Shortlist G8
The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$. Knowing that $ab = \frac{xy}{2}$, show that $\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right)$ or $\left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right)$.
4 replies
parmenides51
Oct 10, 2017
StarLex1
May 6, 2023
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2008 JBMO Shortlist G8
G H J
Reveal topic tags
geometry
JBMO
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G H BBookmark kLocked kLocked NReply
Source: 2008 JBMO Shortlist G8
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parmenides51
30629 posts
parmenides51
#1 h  Oct 10, 2017, 4:33 PM • 2 Y
Y by Adventure10, Mango247
The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$. Knowing that $ab = \frac{xy}{2}$, show that $\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right)$ or $\left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right)$.
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TemetNosce
10 posts
TemetNosce
#2 h  Jun 13, 2022, 5:47 PM • 1 Y
Y by AlperenINAN
Official solution can be found here: https://pregatirematematicaolimpiadejuniori.wordpress.com/2016/07/27/shl-2008/
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Lemmas
11 posts
Lemmas
#3 h  May 5, 2023, 6:27 PM
Y by
You can use 2(a²+b²)=x²+y²
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Almora
41 posts
Almora
#4 h  May 5, 2023, 8:38 PM
Y by
Complex plane suffices. $\blacksquare$
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StarLex1
812 posts
StarLex1
#5 h  May 6, 2023, 11:16 AM
Y by
Stewart
$(\frac{x^2+y^2}{4}) = \frac{a^2+b^2}{2}$
since $ab=\frac{xy}{2}$
$a+b=\frac{x+y}{\sqrt{2}}$
sub $b = \frac{xy}{2a}$
$2a^2+a(-\sqrt{2}x-\sqrt{2}y)+xy=0$
note that (a,b) is symmetric so (a,b) sols to the following quadratic eqn
leading that $(a) = (\frac{x}{\sqrt{2}} ,\frac{y}{\sqrt{2}})$ same wise with b
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