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a My Retirement & New Leadership at AoPS
rrusczyk   1299
N 6 minutes ago by Mbwu
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
1299 replies
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rrusczyk
Monday at 6:37 PM
Mbwu
6 minutes ago
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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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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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Inequalities
sqing   0
4 minutes ago
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +21abc\leq\frac{512}{441}$$Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$$Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$$Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$
0 replies
sqing
4 minutes ago
0 replies
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Prime-related integers [CMO 2018 - P3]
Amir Hossein   15
N 44 minutes ago by Ilikeminecraft
Source: 2018 Canadian Mathematical Olympiad - P3
Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related.

Note that $1$ and $n$ are included as divisors.
15 replies
Amir Hossein
Mar 31, 2018
Ilikeminecraft
44 minutes ago
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Inspired by IMO 1984
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +17abc\leq\frac{8000}{7803}$$$$a^2+b^2+ ab +\frac{163}{10}abc\leq\frac{7189057}{7173630}$$$$a^2+b^2+ ab +16.23442238abc\le1$$
2 replies
1 viewing
sqing
Yesterday at 3:04 PM
sqing
an hour ago
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2025 Caucasus MO Juniors P6
BR1F1SZ   1
N an hour ago by maromex
Source: Caucasus MO
A point $P$ is chosen inside a convex quadrilateral $ABCD$. Could it happen that$$PA = AB, \quad PB = BC, \quad PC = CD \quad \text{and} \quad PD = DA?$$
1 reply
BR1F1SZ
2 hours ago
maromex
an hour ago
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2025 Caucasus MO Juniors P7
BR1F1SZ   2
N an hour ago by doongus
Source: Caucasus MO
It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Could it happen that from segments of lengths $$\sqrt{a^2 + \frac{2}{3} bc},\quad \sqrt{b^2 + \frac{2}{3} ca}\quad \text{and} \quad \sqrt{c^2 + \frac{2}{3} ab},$$a right-angled triangle can be formed?
2 replies
BR1F1SZ
2 hours ago
doongus
an hour ago
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divisors on a circle
Valentin Vornicu   46
N an hour ago by doongus
Source: USAMO 2005, problem 1, Zuming Feng
Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
46 replies
1 viewing
Valentin Vornicu
Apr 21, 2005
doongus
an hour ago
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Maximum of Incenter-triangle
mpcnotnpc   1
N 2 hours ago by mpcnotnpc
Triangle $\Delta ABC$ has side lengths $a$, $b$, and $c$. Select a point $P$ inside $\Delta ABC$, and construct the incenters of $\Delta PAB$, $\Delta PBC$, and $\Delta PAC$ and denote them as $I_A$, $I_B$, $I_C$. What is the maximum area of the triangle $\Delta I_A I_B I_C$?
1 reply
mpcnotnpc
Yesterday at 6:24 PM
mpcnotnpc
2 hours ago
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IMO 2009, Problem 2
orl   141
N 2 hours ago by mananaban
Source: IMO 2009, Problem 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP = OQ.$

Proposed by Sergei Berlov, Russia
141 replies
orl
Jul 15, 2009
mananaban
2 hours ago
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integral points
jhz   0
2 hours ago
Source: 2025 CTST P17
Prove: there exist integer $x_1,x_2,\cdots x_{10},y_1,y_2,\cdots y_{10}$ satisfying the following conditions:
$(1)$ $|x_i|,|y_i|\le 10^{10} $ for all $1\le i \le 10$
$(2)$ Define the set \[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]then \(|S| = 1024\)，and any rectangular strip of width 1 covers at most two points of S.
0 replies
jhz
2 hours ago
0 replies
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Something nice
KhuongTrang   20
N 2 hours ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
20 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
2 hours ago
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2025 Caucasus MO Juniors P5
BR1F1SZ   0
2 hours ago
Source: Caucasus MO
Suppose that $n$ consecutive positive integers were written on the board, where $n > 6$. Then some $5$ of the written numbers were erased, and it turned out that any two of the remaining numbers are coprime. Find the largest possible value of $n$.
0 replies
BR1F1SZ
2 hours ago
0 replies
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best aime/amc10 resources
Spacepandamath13   4
N 3 hours ago by shadow_sensei65283
I want to qual for JMO and watching videos are helpful but only for amc10 (sohil rathi). Anyone have resources for AIME because I'm too broke to afford another AoPS class?
4 replies
Spacepandamath13
Monday at 9:49 PM
shadow_sensei65283
3 hours ago
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digit reversing and divisibility
roundtablepizza   6
N 3 hours ago by roundtablepizza
an interesting problem i thought of:

for what integers k will the following statement be true: if k divides a number, then it will also divide that number reversed.

for example, since 3 divides 321, it also divides 123.

i know this applies for 3, 9, and 11(maybe??) but are there infinitely many more values of k?
6 replies
roundtablepizza
Mar 24, 2025
roundtablepizza
3 hours ago
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ARML math competetion
purpledonutdragon   3
N 5 hours ago by AbhayAttarde01
Do you guys have any tips for ARML? What are some concepts that will be very helpful in ARML?
3 replies
purpledonutdragon
Yesterday at 12:39 PM
AbhayAttarde01
5 hours ago
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that statement is true
pennypc123456789   2
N Mar 23, 2025 by pennypc123456789
we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2$ \le 2 $ ?
2 replies
pennypc123456789
Mar 23, 2025
pennypc123456789
Mar 23, 2025
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that statement is true
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Reveal topic tags
inqualities
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pennypc123456789
16 posts
pennypc123456789
#1 h  Mar 23, 2025, 9:27 AM
Y by
we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2$ \le 2 $ ?
This post has been edited 1 time. Last edited by pennypc123456789, Mar 23, 2025, 10:10 AM
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no_room_for_error
322 posts
no_room_for_error
#2 h  Mar 23, 2025, 11:15 AM • 2 Y
Y by ehuseyinyigit, pennypc123456789
pennypc123456789 wrote:
we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2 \le 2 $ ?

Just $3(a^4+b^4)+2a^4b^4\leq 8$ is enough. It follows from Cauchy-Schwarz

$$25 \geq (2a^4+3)(2b^4+3)\geq (2a^2+2b^2+1)^2$$
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pennypc123456789
16 posts
pennypc123456789
#3 h  Mar 23, 2025, 2:44 PM
Y by
wow thanks
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