Komal A Problems(Hungarian Magazine)

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

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]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies

jlacosta

Mar 2, 2025

0 replies

Komal A727

junioragd 3

N Oct 8, 2023 by ChessMath

A. 727. For any finite sequence $(x_1,…,x_n)$ , denote by $N_{(x_1,…,x_n)}$ the number of ordered index pairs $(i,j)$ for which $1 \le i<j\le n$ and $xi=xj$ . Let $p$ be an odd prime, $1 \le n<p$ , and let $a_1,a_2,…,a_n$ and $b_1,b_2,…,b_n$ be arbitrary residue classes modulo $p$ . Prove that there exists a permutation $q$ of the indices $1,2,…,n$ for which

$N_{(a_1+b_q(1),a_2+b_q(2),…,a_n+b_q(n))}/le min(N_{(a_1,a_2,…,a_n)},N_{(b_1,b_2,…,b_n))}$ .

3 replies

junioragd

Mar 5, 2020

ChessMath

Oct 8, 2023

Komal A709

parmenides51 1

N Oct 7, 2023 by ChessMath

Let $a>0$ be a real number. Find the minimal constant $C_a$ for which the inequality $C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}$ holds for any positive integer $n$ and any sequence $0=x_0<x_1<...<x_n$ of real numbers.

1 reply

parmenides51

Apr 16, 2020

ChessMath

Oct 7, 2023

komal A.776

Joider 0

May 11, 2023

Let $k > 1$ be a fixed odd number, and for non-negative integers $n$ let

$f_n=\sum_{\substack{0\leq i\leq n\\ k\mid n-2i}}\binom{n}{i}.$ Prove that $f_n$ satisfy the following recursion:
$f_{n}^2=\sum_{i=0}^{n} \binom{n}{i}f_{i}f_{n-i}.$

0 replies

Joider

May 11, 2023

0 replies

Komal A703

junioragd 1

N Jan 29, 2023 by CANBANKAN

A. 703. Let $n\ge2$ be an integer. We call an ordered n-tuple of integers primitive if the greatest common divisor of its components is $1$ . Prove that for every finite set $H$ of primitive $n$ -tuples, there exists a non-constant homogenous polynomial $f(x1,x2,…,xn)$ with integer coefficients whose value is $1$ at every $n$ -tuple in $H$ .

1 reply

junioragd

Feb 26, 2020

CANBANKAN

Jan 29, 2023

Komal A754

junioragd 3

N Apr 2, 2022 by Mahdi.sh

A. 754. Let $P$ be a point inside the acute triangle $ABC$ , and let $Q$ be the isogonal conjugate of $P$ . Let $L, M$ and $N$ be the midpoints of the shorter arcs $BC, CA$ and $AB$ of the circumcircle of $ABC$ , respectively. Let $X_A$ be the intersection of ray $LQ$ and circle $PBC$ , let $X_B$ be the intersection of ray $MQ$ and circle $PCA$ , and let $X_C$ be the intersection of ray $NQ$ and circle $PAB$ . Prove that $P, X_A, X_B$ and $X_C$ are concyclic or coincide.

3 replies

junioragd

Feb 25, 2020

Mahdi.sh

Apr 2, 2022

Komal A725

junioragd 1

N Mar 19, 2022 by parmenides51

A. 725. Let $R+$ denote the set of positive real numbers. Find all functions $f:R+/rightarrowR+$ satisfying the following equation for all $x,y$ in $R+$ :

$f(xy+f(y)^2)=f(x)f(y)+yf(y)$ .

1 reply

junioragd

Mar 5, 2020

parmenides51

Mar 19, 2022

Komal A751

junioragd 1

N Mar 19, 2022 by parmenides51

A. 751. Let $c>0$ be a real number, and suppose that for every positive integer $n$ , at least one percent of the numbers $1^c,2^c,3^c,…,n^c$ are integers. Prove that $c$ is an integer.

1 reply

junioragd

Feb 25, 2020

parmenides51

Mar 19, 2022

Komal A post collections

parmenides51 1

N Mar 19, 2022 by parmenides51

Komal A.701-754

1 reply

parmenides51

Apr 12, 2020

parmenides51

Mar 19, 2022

Komal A743

junioragd 1

N Jul 21, 2021 by iman007

A. 743. The incircle of tangential quadrilateral $ABCD$ intersects diagonal $BD$ at $P$ and $Q$ ( $BP<BQ$ ). Let $UV$ be the diameter of the incircle perpendicular to $AC$ ( $BU \le BV$ ). Show that the lines $AC, PV$ and $QU$ pass through one point.

1 reply

junioragd

Feb 25, 2020

iman007

Jul 21, 2021

Komal A705

junioragd 1

N Oct 31, 2020 by parmenides51

A. 705. Triangle $ABC$ has orthocenter $H$ . Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$ . Suppose that circle $BHD$ meets $AB$ at $P!=B$ , and circle $CHD$ meets $AC$ at $Q!=C$ . Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle.

1 reply

junioragd

Feb 26, 2020

parmenides51

Oct 31, 2020