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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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A number theory problem from the British Math Olympiad
Rainbow1971   12
N 9 minutes ago by ektorasmiliotis
Source: British Math Olympiad, 2006/2007, round 1, problem 6
I am a little surprised to find that I am (so far) unable to solve this little problem:

[quote]Let $n$ be an integer. Show that, if $2 + 2 \sqrt{1+12n^2}$ is an integer, then it is a perfect square.[/quote]

I set $k := \sqrt{1+12n^2}$. If $2 + 2 \sqrt{1+12n^2}$ is an integer, then $k (=\sqrt{1+12n^2})$ is at least rational, so that $1 + 12n^2$ must be a perfect square then. Using Conway's topograph method, I have found out that the smallest non-negative pairs $(n, k)$ for which this happens are $(0,1), (2,7), (28,97)$ and $(390, 1351)$, and that, for every such pair $(n,k)$, the "next" such pair can be calculated as
$$ \begin{bmatrix} 7 & 2 \\ 24 & 7 \end{bmatrix} \begin{bmatrix} n \\ k \end{bmatrix} .$$The eigenvalues of that matrix are irrational, however, so that any calculation which uses powers of that matrix is a little cumbersome. There must be an easier way, but I cannot find it. Can you?

Thank you.




12 replies
Rainbow1971
Yesterday at 8:39 PM
ektorasmiliotis
9 minutes ago
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A number theory about divisors which no one fully solved at the contest
nAalniaOMliO   22
N 12 minutes ago by nAalniaOMliO
Source: Belarusian national olympiad 2024
Let's call a pair of positive integers $(k,n)$ interesting if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$
Find the number of interesting pairs $(k,n)$ with $k \leq 100$
M. Karpuk
22 replies
1 viewing
nAalniaOMliO
Jul 24, 2024
nAalniaOMliO
12 minutes ago
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D1018 : Can you do that ?
Dattier   1
N 16 minutes ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
16 minutes ago
J
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Nordic 2025 P3
anirbanbz   8
N an hour ago by Primeniyazidayi
Source: Nordic 2025
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$.
8 replies
anirbanbz
Mar 25, 2025
Primeniyazidayi
an hour ago
J
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2022 olympiad geometry (euclidean & combo) from contests
parmenides51   3
N Nov 9, 2022 by parmenides51
Source: geometry problems from 2022 contest, classic Euclidean mostly
I have started collecting all the geometry problems posted in aops from 2022 in a forum collection.

Here is that forum collection, that contains any collection mentioned below.

so far I have added geometry problems from:
$\bullet$ 2022 International Contests
$\bullet$ 2021 International Shortlists (IMO + BMO + JBMO)
$\bullet$ 2022 National Contests / America [Seniors + Juniors]
$\bullet$ 2022 National Contests / Asia + Oceania [Seniors + Juniors]
$\bullet$ 2022 National Contests / Europe + Africa [Seniors + Juniors]
$\bullet$ 2022 Geometry Olympiads [Oral Moscow, Sharygin, Yasinksy]
$\bullet$ 2022 Regional Contests
$\bullet$ 2022 TSTs (not from Shortlists) [IMO + EGMO + Ibero + Cono Sur]
$\bullet$ 2022 JBMO TST
$\bullet$ 2022 Unofficial Contests
$\bullet$ 2022 Mocks

Related is also 2022 -Combo Geo ,
which includes combinatorial geometry problems (without tiles nor grid) from
$\bullet$ 2022 International Contests
$\bullet$ 2022 National Contests
$\bullet$ 2022 Regional Contests
$\bullet$ 2022 TSTs (not from Shortlists)

enjoy / start solving
3 replies
parmenides51
Sep 21, 2022
parmenides51
Nov 9, 2022
J
2022 olympiad geometry (euclidean & combo) from contests
G H J
Reveal topic tags
Isometric axis
L
G H BBookmark kLocked kLocked NReply
Source: geometry problems from 2022 contest, classic Euclidean mostly
The post below has been deleted. Click to close.
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parmenides51
30628 posts
parmenides51
#1 h  Sep 21, 2022, 1:33 PM
Y by
I have started collecting all the geometry problems posted in aops from 2022 in a forum collection.

Here is that forum collection, that contains any collection mentioned below.

so far I have added geometry problems from:
$\bullet$ 2022 International Contests
$\bullet$ 2021 International Shortlists (IMO + BMO + JBMO)
$\bullet$ 2022 National Contests / America [Seniors + Juniors]
$\bullet$ 2022 National Contests / Asia + Oceania [Seniors + Juniors]
$\bullet$ 2022 National Contests / Europe + Africa [Seniors + Juniors]
$\bullet$ 2022 Geometry Olympiads [Oral Moscow, Sharygin, Yasinksy]
$\bullet$ 2022 Regional Contests
$\bullet$ 2022 TSTs (not from Shortlists) [IMO + EGMO + Ibero + Cono Sur]
$\bullet$ 2022 JBMO TST
$\bullet$ 2022 Unofficial Contests
$\bullet$ 2022 Mocks

Related is also 2022 -Combo Geo ,
which includes combinatorial geometry problems (without tiles nor grid) from
$\bullet$ 2022 International Contests
$\bullet$ 2022 National Contests
$\bullet$ 2022 Regional Contests
$\bullet$ 2022 TSTs (not from Shortlists)

enjoy / start solving
This post has been edited 47 times. Last edited by parmenides51, Nov 16, 2022, 3:23 PM
Z K Y
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parmenides51
30628 posts
parmenides51
#3 h  Oct 31, 2022, 2:42 PM
Y by
many recent problems have been added

enjoy / start solving
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
parmenides51
30628 posts
parmenides51
#4 h  Nov 3, 2022, 7:19 PM
Y by
just added 2022 JBMO TST / geometry above
Z K Y
The post below has been deleted. Click to close.
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parmenides51
30628 posts
parmenides51
#5 h  Nov 9, 2022, 3:01 AM
Y by
just added 2022 Geometry Olympiads [Oral Moscow, Sharygin, Yasinksy]
Z K Y
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