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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]
Our full course list for upcoming classes is below:
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0 replies
jlacosta
Mar 2, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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VERY HARD MATH PROBLEM!
slimshadyyy.3.60   15
N 20 minutes ago by Burmf
Let a ≥b ≥c ≥0 be real numbers such that a^2 +b^2 +c^2 +abc = 4. Prove that
a+b+c+(√a−√c)^2 ≥3.
15 replies
2 viewing
slimshadyyy.3.60
Yesterday at 10:49 PM
Burmf
20 minutes ago
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Midpoints of chords on a circle
AwesomeToad   38
N 26 minutes ago by LeYohan
Source: 0
Let $C$ be a circle and $P$ a given point in the plane. Each line through $P$ which intersects $C$ determines a chord of $C$. Show that the midpoints of these chords lie on a circle.
38 replies
AwesomeToad
Sep 23, 2011
LeYohan
26 minutes ago
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Polish MO finals, problem 1
michaj   4
N 32 minutes ago by AshAuktober
In each cell of a matrix $ n\times n$ a number from a set $ \{1,2,\ldots,n^2\}$ is written --- in the first row numbers $ 1,2,\ldots,n$, in the second $ n+1,n+2,\ldots,2n$ and so on. Exactly $ n$ of them have been chosen, no two from the same row or the same column. Let us denote by $ a_i$ a number chosen from row number $ i$. Show that:

\[ \frac{1^2}{a_1}+\frac{2^2}{a_2}+\ldots +\frac{n^2}{a_n}\geq \frac{n+2}{2}-\frac{1}{n^2+1}\]
4 replies
michaj
Apr 10, 2008
AshAuktober
32 minutes ago
J
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2025 Caucasus MO Seniors P7
BR1F1SZ   1
N 38 minutes ago by X.Luser
Source: Caucasus MO
From a point $O$ lying outside the circle $\omega$, two tangents are drawn touching $\omega$ at points $M$ and $N$. A point $K$ is chosen on the segment $MN$. Let points $P$ and $Q$ be the midpoints of segments $KM$ and $OM$ respectively. The circumcircle of triangle $MPQ$ intersects $\omega$ again at point $L$ ($L \neq M$). Prove that the line $LN$ passes through the centroid of triangle $KMO$.
1 reply
BR1F1SZ
Mar 26, 2025
X.Luser
38 minutes ago
J
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Easy geometry
Bluesoul   13
N 39 minutes ago by AshAuktober
Source: CJMO 2022 P1
Let $\triangle{ABC}$ has circumcircle $\Gamma$, drop the perpendicular line from $A$ to $BC$ and meet $\Gamma$ at point $D$, similarly, altitude from $B$ to $AC$ meets $\Gamma$ at $E$. Prove that if $AB=DE, \angle{ACB}=60^{\circ}$
(sorry it is from my memory I can't remember the exact problem, but it means the same)
13 replies
1 viewing
Bluesoul
Mar 12, 2022
AshAuktober
39 minutes ago
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IMO Shortlist 2013, Geometry #2
lyukhson   77
N an hour ago by endless_abyss
Source: IMO Shortlist 2013, Geometry #2
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
77 replies
lyukhson
Jul 9, 2014
endless_abyss
an hour ago
J
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f(x*f(y)) = f(x)/y
orl   23
N an hour ago by Maximilian113
Source: IMO 1990, Day 2, Problem 4, IMO ShortList 1990, Problem 25 (TUR 4)
Let $ {\mathbb Q}^ +$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ + \rightarrow {\mathbb Q}^ +$ such that
\[ f(xf(y)) = \frac {f(x)}{y} \]
for all $ x$, $ y$ in $ {\mathbb Q}^ +$.
23 replies
orl
Nov 11, 2005
Maximilian113
an hour ago
J
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Heavy config geo involving mixtilinear
Assassino9931   2
N an hour ago by Assassino9931
Source: Bulgaria Spring Mathematical Competition 2025 12.4
Let $ABC$ be an acute-angled triangle \( ABC \) with \( AC > BC \) and incenter \( I \). Let \( \omega \) be the mixtilinear circle at vertex \( C \), i.e. the circle internally tangent to the circumcircle of \( \triangle ABC \) and also tangent to lines \( AC \) and \( BC \). A circle \( \Gamma \) passes through points \( A \) and \( B \) and is tangent to \( \omega \) at point \( T \), with \( C \notin \Gamma \) and \( I \) being inside \( \triangle ATB \). Prove that:
$$\angle CTB + \angle ATI = 180^\circ + \angle BAI - \angle ABI.$$
2 replies
Assassino9931
6 hours ago
Assassino9931
an hour ago
J
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Guess the leader's binary string!
cjquines0   78
N an hour ago by de-Kirschbaum
Source: 2016 IMO Shortlist C1
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
78 replies
cjquines0
Jul 19, 2017
de-Kirschbaum
an hour ago
J
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Monkeys have bananas
nAalniaOMliO   5
N an hour ago by jkim0656
Source: Belarusian National Olympiad 2025
Ten monkeys have 60 bananas. Each monkey has at least one banana and any two monkeys have different amounts of bananas.
Prove that any six monkeys can distribute their bananas between others such that all 4 remaining monkeys have the same amount of bananas.
5 replies
1 viewing
nAalniaOMliO
Friday at 8:20 PM
jkim0656
an hour ago
J
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Fixed point config on external similar isosceles triangles
Assassino9931   1
N an hour ago by E50
Source: Bulgaria Spring Mathematical Competition 2025 10.2
Let $AB$ be an acute scalene triangle. A point \( D \) varies on its side \( BC \). The points \( P \) and \( Q \) are the midpoints of the arcs \( \widehat{AB} \) and \( \widehat{AC} \) (not containing \( D \)) of the circumcircles of triangles \( ABD \) and \( ACD \), respectively. Prove that the circumcircle of triangle \( PQD \) passes through a fixed point, independent of the choice of \( D \) on \( BC \).
1 reply
Assassino9931
Today at 12:41 PM
E50
an hour ago
J
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Problem 2
Functional_equation   15
N an hour ago by basilis
Source: Azerbaijan third round 2020
$a,b,c$ are positive integer.
Solve the equation:
$ 2^{a!}+2^{b!}=c^3 $
15 replies
Functional_equation
Jun 6, 2020
basilis
an hour ago
J
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Intersection of a cevian with the incircle
djb86   24
N 2 hours ago by Ilikeminecraft
Source: South African MO 2005 Q4
The inscribed circle of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. Let $Q$ denote the other point of intersection of $AD$ and the inscribed circle. Prove that $EQ$ extended passes through the midpoint of $AF$ if and only if $AC = BC$.
24 replies
djb86
May 27, 2012
Ilikeminecraft
2 hours ago
J
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Polynomials and their shift with all real roots and in common
Assassino9931   2
N 2 hours ago by AshAuktober
Source: Bulgaria Spring Mathematical Competition 2025 11.4
We call two non-constant polynomials friendly if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials \( P(x), Q(x) \) and a constant \( C \in \mathbb{R}, C \neq 0 \), it is given that \( P(x) + C \) and \( Q(x) + C \) are also friendly polynomials. Prove that \( P(x) \equiv Q(x) \).
2 replies
Assassino9931
6 hours ago
AshAuktober
2 hours ago
J
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J
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Cauchy-Schwarz 2
prtoi   5
N Mar 27, 2025 by Filipjack
Source: Handout by Samin Riasat
if $a^2+b^2+c^2+d^2=4$, prove that:
$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\ge4$
5 replies
prtoi
Mar 26, 2025
Filipjack
Mar 27, 2025
J
Cauchy-Schwarz 2
G H J
Reveal topic tags
inequalities
L
G H BBookmark kLocked kLocked NReply
Source: Handout by Samin Riasat
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prtoi
37 posts
prtoi
#1 h  Mar 26, 2025, 4:19 PM • 2 Y
Y by teomihai, MathPerson12321
if $a^2+b^2+c^2+d^2=4$, prove that:
$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\ge4$
Z K Y
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Primeniyazidayi
28 posts
Primeniyazidayi
#2 h  Mar 26, 2025, 6:21 PM
Y by
Hint:Titu's Lemma
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mpcnotnpc
44 posts
mpcnotnpc
#3 h  Mar 26, 2025, 7:51 PM
Y by
Click to reveal hidden text
This post has been edited 1 time. Last edited by mpcnotnpc, Mar 26, 2025, 7:51 PM
Z K Y
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lbh_qys
450 posts
lbh_qys
#4 h  Mar 27, 2025, 2:56 AM
Y by
According to Hölder's inequality,

\[ \left(\sum \frac{a^2}{b}\right)^2 \left(\sum a^2b^2\right) \ge \left(\sum a^2\right)^3 = 4^3 \]
and

\[ \sum a^2b^2 = (a^2+c^2)(b^2+d^2) \le \frac{\left(\sum a^2\right)^2}{4} = 4. \]
Hence,

\[ \sum \frac{a^2}{b} \ge \sqrt{\frac{4^3}{4}} = 4. \]
Z K Y
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sqing
41325 posts
sqing
#5 h  Mar 27, 2025, 3:05 AM
Y by
prtoi wrote:
if $a^2+b^2+c^2+d^2=4$, prove that:
$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\ge4$
https://artofproblemsolving.com/community/c6h212248p1170886
Z K Y
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Filipjack
827 posts
Filipjack
#6 h  Mar 27, 2025, 10:46 AM
Y by
mpcnotnpc wrote:
Click to reveal hidden text

How do you get that from Titu?
Z K Y
N Quick Reply
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a