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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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H
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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2016 PUMaC Individual Finals A3
parmenides51   7
N Yesterday at 12:23 AM by lpieleanu
On a cyclic quadrilateral $ABCD$, $M$ is the midpoint of $AB$ and $N$ is the midpoint of $CD$. Let $E$ be the projection of $C$ onto $AB$ and $F$ the reflection of $N$ about the midpoint of $DE$. If $F$ is inside quadrilateral $ABCD$, show that $\angle BMF = \angle CBD$.
7 replies
parmenides51
Sep 22, 2019
lpieleanu
Yesterday at 12:23 AM
J
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Finals 2013 A3: An edge matching
v_Enhance   21
N Mar 21, 2025 by ihategeo_1969
A graph consists of a set of vertices, some of which are connected by (undirected) edges. A star of a graph is a set of edges with a common endpoint. A matching of a graph is a set of edges such that no two have a common endpoint. Show that if the number of edges of a graph $G$ is larger than $2(k-1)^2$, then $G$ contains a matching of size $k$ or a star of size $k$.
21 replies
v_Enhance
Nov 17, 2013
ihategeo_1969
Mar 21, 2025
J
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2019 PUMaC Individual Finals A1
parmenides51   12
N Feb 23, 2025 by quantam13
Given the graph $G$ and cycle $C$ in it, we can perform the following operation: add another vertex $v$ to the graph, connect it to all vertices in $C$ and erase all the edges from $C$. Prove that we cannot perform the operation indefinitely on a given graph.
12 replies
parmenides51
Dec 26, 2021
quantam13
Feb 23, 2025
J
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2016 PUMaC Number Theory A7 / B8
parmenides51   13
N Feb 12, 2025 by ryanbear
Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1$ (mod $2017$). ($2017$ is prime.)
13 replies
parmenides51
Sep 22, 2019
ryanbear
Feb 12, 2025
J
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2018 PUMaC Number Theory A5
mathman3880   7
N Jan 25, 2025 by EVS383
Find the remainder when
$$\prod_{i = 1}^{1903} (2^i + 5)$$is divided by $1000$.
7 replies
mathman3880
Nov 25, 2018
EVS383
Jan 25, 2025
J
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2023 PUMaC Algebra B2
lpieleanu   2
N Jan 20, 2025 by P162008
The sum $$\sum_{m=1}^{2023} \frac{2m}{m^4+m^2+1}$$can be expressed as $\tfrac{a}{b}$ for relatively prime positive integers $a,b.$ Find the remainder when $a+b$ is divided by $1000.$
2 replies
lpieleanu
Oct 5, 2024
P162008
Jan 20, 2025
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2024 PUMaC Algebra B1
lpieleanu   3
N Jan 12, 2025 by m64sky
Let $A=\sqrt{7+2\sqrt{10}} - \sqrt{7-2\sqrt{10}}.$ We can express $A$ as $a\sqrt{b},$ where $a,b$ are integers and $b$ is square-free. Compute $a+b.$
3 replies
lpieleanu
Nov 24, 2024
m64sky
Jan 12, 2025
J
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2024 PUMaC Algebra A7
lpieleanu   1
N Jan 2, 2025 by yorufan
Let $F_1=1, F_2=1,$ and $F_{n+2}=F_{n+1}+F_n.$ Then, $$S = \sum_{n=1}^{\infty} \arctan\left(\frac{1}{F_n}\right)\arctan\left(\frac{1}{F_{n+1}}\right)$$Find $\lfloor 80S \rfloor.$

(Hint: it may be useful to note that $\arctan(\tfrac{1}{1}) = \arctan(\tfrac{1}{2})+\arctan(\tfrac{1}{3}).$)
1 reply
lpieleanu
Nov 25, 2024
yorufan
Jan 2, 2025
J
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2024 PUMaC Number Theory A5 / B7
lpieleanu   3
N Dec 22, 2024 by Mr.Sharkman
Call a positive integer nice if the sum of its even proper divisors is larger than the sum of its odd proper divisors. What is the smallest nice number that is congruent to $2 \text{ mod } 4$?
3 replies
lpieleanu
Nov 25, 2024
Mr.Sharkman
Dec 22, 2024
J
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2024 PUMaC Number Theory A4 / B6
lpieleanu   2
N Dec 22, 2024 by Mr.Sharkman
Let $r(m)$ be the number of positive integers a less than or equal to $m$ where $\gcd(a, m)$ is prime. Find the sum of all positive integers $m < 300$ such that $r(m) = \varphi(m),$ where $\varphi(m)$ denotes the number of positive integers $a$ less than $m$ where $\gcd(a, m) = 1.$
2 replies
lpieleanu
Nov 25, 2024
Mr.Sharkman
Dec 22, 2024
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J
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2023 PUMaC Algebra B2
lpieleanu   2
N Jan 20, 2025 by P162008
The sum $$\sum_{m=1}^{2023} \frac{2m}{m^4+m^2+1}$$can be expressed as $\tfrac{a}{b}$ for relatively prime positive integers $a,b.$ Find the remainder when $a+b$ is divided by $1000.$
2 replies
lpieleanu
Oct 5, 2024
P162008
Jan 20, 2025
J
2023 PUMaC Algebra B2
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lpieleanu
2833 posts
lpieleanu
#1 h  Oct 5, 2024, 3:16 PM
Y by
The sum $$\sum_{m=1}^{2023} \frac{2m}{m^4+m^2+1}$$can be expressed as $\tfrac{a}{b}$ for relatively prime positive integers $a,b.$ Find the remainder when $a+b$ is divided by $1000.$
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lpieleanu
2833 posts
lpieleanu
#2 h  Oct 7, 2024, 12:38 AM
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Solution
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P162008
94 posts
P162008
#3 h  Jan 20, 2025, 11:19 PM
Y by
Dr = m⁴ + m² + 1 = (m² + m + 1)(m² - m + 1)

Nr = 2m = (m² + m + 1) - (m² - m + 1)

Now, S = 1/(m² - m + 1) - 1/(m² + m + 1) which telescopes to 1 - 1/(2023² + 2023 + 1)

a/b = (2023² + 2023)/(2023² + 2024)

Therefore, a + b(mod 1000) = 2.2023² + 47 = 58 + 47 = 105(mod 1000)
This post has been edited 2 times. Last edited by P162008, Jan 22, 2025, 10:01 AM
Reason: Typo
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