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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 functional equation from MEMO
square_root_of_3   24
N 3 minutes ago by pco
Source: Middle European Mathematical Olympiad 2022, problem I-1
Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(x+f(x+y))=x+f(f(x)+y)$$holds for all real numbers $x$ and $y$.
24 replies
square_root_of_3
Sep 1, 2022
pco
3 minutes ago
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Functional equations
hanzo.ei   0
6 minutes ago
Source: Greekldiot
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
0 replies
hanzo.ei
6 minutes ago
0 replies
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Not so classic orthocenter problem
m4thbl3nd3r   4
N 15 minutes ago by hanzo.ei
Source: own?
Let $O$ be circumcenter of a non-isosceles triangle $ABC$ and $H$ be a point in the interior of $\triangle ABC$. Let $E,F$ be foots of perpendicular lines from $H$ to $AC,AB$. Suppose that $BCEF$ is cyclic and $M$ is the circumcenter of $BCEF$, $HM\cap AB=K,AO\cap BE=T$. Prove that $KT$ bisects $EF$
4 replies
m4thbl3nd3r
Yesterday at 4:59 PM
hanzo.ei
15 minutes ago
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Numbers not power of 5
Kayak   33
N 24 minutes ago by ihategeo_1969
Source: Indian TST D1 P2
Show that there do not exist natural numbers $a_1, a_2, \dots, a_{2018}$ such that the numbers \[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]are all powers of $5$

Proposed by Tejaswi Navilarekallu
33 replies
Kayak
Jul 17, 2019
ihategeo_1969
24 minutes ago
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Conics Problem
Saucepan_man02   1
N 3 hours ago by vanstraelen
Let the asymptotes of a hyperbola be $3x-2y-1=0$ and $2x-3y+5=0$ and one of its tangents be $x=y$. Find the square of transverse axis.
1 reply
Saucepan_man02
4 hours ago
vanstraelen
3 hours ago
J
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functions false or true
Math2030   8
N 3 hours ago by Mathzeus1024
find all functions f: \mathbb{R} \to \mathbb{R} that satisfy the functional equation:


f(x^2 f(x) + f(y)) = (f(x))^3 + f(y), \quad \forall x, y \in \mathbb{R}
8 replies
Math2030
Mar 25, 2025
Mathzeus1024
3 hours ago
J
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An inequality
JK1603JK   0
4 hours ago
Let a,b,c\ge 0: ab+bc+ca>0 then prove \frac{5ab+c^2}{a+b}+\frac{5bc+a^2}{b+c}+\frac{5ca+b^2}{c+a}\ge 9\cdot\frac{ab+bc+ca}{a+b+c}.
0 replies
JK1603JK
4 hours ago
0 replies
J
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Inequalities
sqing   11
N 4 hours ago by sqing
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a(b+c+ 5bc +1)\leq\frac{676}{675}$$$$a(b+c+6bc +1)\leq\frac{245}{243}$$
11 replies
sqing
Mar 26, 2025
sqing
4 hours ago
J
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IOQM P16 2024
SomeonecoolLovesMaths   3
N Today at 8:59 AM by ohnm
Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a function satisfying the relation $4f(3-x) + 3f(x) = x^2$ for any real $x$. Find the value of $f(27) - f(25)$ to the nearest integer. (Here $\mathbb{R}$ denotes the set of real numbers.)
3 replies
SomeonecoolLovesMaths
Sep 8, 2024
ohnm
Today at 8:59 AM
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Geometry Anticenter
Doanh   0
Today at 7:31 AM
Given an acute, non-isosceles triangle \( \triangle ABC \).
Let \( D \) and \( E \) be points on sides \( AB \) and \( AC \), respectively, such that \( DE \parallel BC \).

Denote \( O' \) and \( O'' \) as the centers of the circumcircles of \( \triangle ABE \) and \( \triangle ACD \), respectively.
The line \( O'O'' \) intersects \( AB \) and \( AC \) at two distinct points \( P \) and \( Q \).

Let \( O \) be the circumcenter of \( \triangle APQ \).
Given that all these points are distinct, prove that the lines \( OA \), \( BE \), and \( CD \) are concurrent at a single point.
0 replies
Doanh
Today at 7:31 AM
0 replies
J
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inequality
JK1603JK   1
N Today at 5:53 AM by aidan0626
Let a,b,c\ge 0: ab+bc+ca>0 then prove \frac{3a-b-c}{b^2+c^2}+\frac{3b-c-a}{c^2+a^2}+\frac{3c-a-b}{a^2+b^2}\ge \frac{3}{2}\cdot\frac{a+b+c}{ab+bc+ca}
1 reply
JK1603JK
Today at 5:50 AM
aidan0626
Today at 5:53 AM
J
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Inequalities
sqing   1
N Today at 5:46 AM by sqing
Let $ a,b,c\geq 1 $ and $ abc-\frac{1}{3}( ab+bc+ca)\leq 4. $ Prove that
$$20\geq 4(a+b+c) - (a b+b c+c a)-a b c \geq 4$$
1 reply
sqing
Mar 27, 2025
sqing
Today at 5:46 AM
J
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Thanks for your help
CHENGQIYU   0
Today at 4:36 AM
There are one circle.
Two points (A,B)out of the circle.
Can you find two lines (AC,CB)
Let AC+CB min
Ps.point C is on the circle
Thanks very much
0 replies
CHENGQIYU
Today at 4:36 AM
0 replies
J
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Good Functional equation question
vexploresmathysics   4
N Today at 3:59 AM by jasperE3
If f : R^+ --> R^+ satisfying f(f(x)/y ) = yf ( y ) + (f(x)). Then the value of α such that Sigma K = 1 to n [ 1 / f(K) ] = 420
4 replies
vexploresmathysics
Jul 1, 2024
jasperE3
Today at 3:59 AM
J
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J
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Smallest value of |253^m - 40^n|
MS_Kekas   3
N Mar 26, 2025 by imagien_bad
Source: Kyiv City MO 2024 Round 1, Problem 9.5
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.

Proposed by Oleksii Masalitin
3 replies
MS_Kekas
Jan 28, 2024
imagien_bad
Mar 26, 2025
J
Smallest value of |253^m - 40^n|
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Reveal topic tags
number theory
Diophantine equation
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G H BBookmark kLocked kLocked NReply
Source: Kyiv City MO 2024 Round 1, Problem 9.5
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MS_Kekas
275 posts
MS_Kekas
#1 h  Jan 28, 2024, 9:35 PM • 1 Y
Y by Tintarn
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.

Proposed by Oleksii Masalitin
Z K Y
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Tintarn
9029 posts
Tintarn
#2 h  Jan 28, 2024, 11:02 PM • 1 Y
Y by Pal702004
Solution
Z K Y
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thewizard369
15 posts
thewizard369
#3 h  Feb 11, 2024, 7:24 PM
Y by
same solution 9 is the minimal value
Z K Y
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imagien_bad
36 posts
imagien_bad
#4 h  Mar 26, 2025, 3:33 AM
Y by
ans is 9 which is ook by 253^2 - 40^3 now ftsofc <9

by mod 2 its odd and by mod 3 its diivs by 3 so must be 3

by mod 8 253^m-40^n is 5 or 1 mod 8 if m is odd or even but 1 mod 8 is bad so must be 5 mod 8 and equal -3 and m odd

then by mod 7 40^n = 4 mod 7 so n = 2 mod 6 but then 40^n mod 9 is 7 which is bad cuz 253^m - 40^n would be -6 mod 9 but -3 is 6 mod 9
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