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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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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 hard
Cobedangiu   2
N 24 minutes ago by IceyCold
problem
2 replies
+1 w
Cobedangiu
Mar 31, 2025
IceyCold
24 minutes ago
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Find the probability
ali3985   0
33 minutes ago
Let $A$ be a set of Natural numbers from $1$ to $N$.
Now choose $k$ ($k \geq 3$) distinct elements from this set.

What is the probability of these numbers to be an increasing geometric progression ?
0 replies
ali3985
33 minutes ago
0 replies
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IMOC 2017 G2 , (ABC) <= (DEF) . perpendiculars related
parmenides51   7
N 38 minutes ago by AshAuktober
Source: https://artofproblemsolving.com/community/c6h1740077p11309077
Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$
7 replies
parmenides51
Mar 20, 2020
AshAuktober
38 minutes ago
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The Sums of Elements in Subsets
bobaboby1   3
N 38 minutes ago by bobaboby1
Given a finite set \( X = \{x_1, x_2, \ldots, x_n\} \), and the pairwise comparison of the sums of elements of all its subsets (with the empty set defined as having a sum of 0), which amounts to \( \binom{2}{2^n} \) inequalities, these given comparisons satisfy the following three constraints:

1. The sum of elements of any non-empty subset is greater than 0.
2. For any two subsets, removing or adding the same elements does not change their comparison of the sums of elements.
3. For any two disjoint subsets \( A \) and \( B \), if the sums of elements of \( A \) and \( B \) are greater than those of subsets \( C \) and \( D \) respectively, then the sum of elements of the union \( A \cup B \) is greater than that of \( C \cup D \).

The question is: Does there necessarily exist a positive solution \( (x_1, x_2, \ldots, x_n) \) that satisfies all these conditions?
3 replies
bobaboby1
Mar 12, 2025
bobaboby1
38 minutes ago
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3 var inquality
sqing   1
N an hour ago by sqing
Source: Own
Let $a,b,c>0$. Prove that
$$ \dfrac{ab}{a^2-ab+3b^2} + \dfrac{bc}{b^2-bc+3c^2} + \dfrac{ca}{c^2-ca+3a^2} \le1$$$$ \dfrac{ab}{a^2-ab+ 2b^2} + \dfrac{bc}{b^2-bc+2 c^2} + \dfrac{ca}{c^2-ca+ 2a^2}\le \dfrac{3}{2}$$$$ \dfrac{ab}{2a^2-ab+3b^2} + \dfrac{bc}{2b^2-bc+3c^2} + \dfrac{ca}{2c^2-ca+3a^2} \le \dfrac{3}{4}$$
1 reply
sqing
an hour ago
sqing
an hour ago
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Show that AB/AC=BF/FC
syk0526   75
N an hour ago by AshAuktober
Source: APMO 2012 #4
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.

(Here we denote $XY$ the length of the line segment $XY$.)
75 replies
syk0526
Apr 2, 2012
AshAuktober
an hour ago
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Very hard FE problem
steven_zhang123   1
N an hour ago by GreekIdiot
Source: 0
Given a real number \(C\) such that \(x + y + z = C\) (where \(x, y, z \in \mathbb{R}\)), and a functional equation \(f: \mathbb{R} \rightarrow \mathbb{R}\) that satisfies \((f^x(y) + f^y(z) + f^z(x))((f(x))^y + (f(y))^z + (f(z))^x) \geq 2025\) for all \(x, y, z \in \mathbb{R}\), has a finite number of solutions. Find such \(C\).
(Here, $f^{n}(x)$ is the function obtained by composing $f(x)$ $n$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{n \ \text{times}})(x)$)
1 reply
steven_zhang123
Mar 30, 2025
GreekIdiot
an hour ago
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inequalities
Cobedangiu   8
N an hour ago by xytunghoanh
problem
8 replies
1 viewing
Cobedangiu
Mar 31, 2025
xytunghoanh
an hour ago
J
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April Fools Geometry
awesomeming327.   4
N an hour ago by avinashp
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ be the projection from $A$ onto $BC$. Let $E$ be a point on the extension of $AD$ past $D$ such that $\angle BAC+\angle BEC=90^\circ$. Let $L$ be on the perpendicular bisector of $AE$ such that $L$ and $C$ are on the same side of $AE$ and
\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]Let the reflection of $D$ across $AB$ and $AC$ be $W$ and $Y$, respectively. Let $X\in AW$ and $Z\in AY$ such that $\angle XBE=\angle ZCE=90^\circ$. Let $EX$ and $EZ$ intersect the circumcircles of $EBD$ and $ECD$ at $J$ and $K$, respectively. Let $LB$ and $LC$ intersect $WJ$ and $YK$ at $P$ and $Q$. Let $PQ$ intersect $BC$ at $F$. Prove that $FB/FC=DB/DC$.
4 replies
awesomeming327.
Yesterday at 2:52 PM
avinashp
an hour ago
J
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Olympiad Geometry problem-second time posting
kjhgyuio   4
N an hour ago by ND_
Source: smo problem
In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
4 replies
kjhgyuio
Today at 1:03 AM
ND_
an hour ago
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Prove that there are no tuples $(x, y, z)$ sastifying $x^2+y^2-z^2=xyz-2$
Anabcde   0
2 hours ago
Prove that there are no tuples $(x, y, z) \in \mathbb{Z}^3$ sastifying $x^2+y^2-z^2=xyz-2$
0 replies
Anabcde
2 hours ago
0 replies
J
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Finding pairs of complex numbers with a certain property
Ciobi_   0
2 hours ago
Source: Romania NMO 2025 10.4
Find all pairs of complex numbers $(z,w) \in \mathbb{C}^2$ such that the relation \[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]holds for all positive integers $n$.
0 replies
Ciobi_
2 hours ago
0 replies
J
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Writing x^k as a product of injective functions
Ciobi_   0
2 hours ago
Source: Romania NMO 2025 10.3
Define the functions $g_k \colon \mathbb{Z} \to \mathbb{Z}$, $g_k(x) = x^k$, where $k$ is a positive integer.
Find the set $M_k$ of positive integers $n$ for which there exist injective functions $f_1,f_2, \dots ,f_n \colon \mathbb{Z} \to \mathbb{Z}$ such that $g_k=f_1\cdot f_2 \cdot \ldots \cdot f_n$.
(Here, $\cdot$ denotes component-wise function multiplication)
0 replies
Ciobi_
2 hours ago
0 replies
J
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2 var inquality
sqing   0
2 hours ago
Source: Own
Let $ a,b \ge 0 $ and $ a+b=2. $ Prove that
$$\sqrt{ a^2+b+6}+\sqrt{ b^2+a+6}\leq 8\sqrt{\frac{2- ab}{ab+1}} $$$$\sqrt{2a^2+b+1}+\sqrt{2b^2+a+1}\leq 4\sqrt{\frac{5-2ab}{ab+2}} $$$$\sqrt{2a^2+b}+\sqrt{2b^2+a}\leq 2\sqrt{\frac{3(5-2ab)}{ab+2}} $$
0 replies
sqing
2 hours ago
0 replies
J
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J
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Geometric inequality
pggp   2
N Apr 8, 2021 by parmenides51
Source: Polish Junior Math Olympiad 2020 2nd Round
Let $ABC$ be such a triangle that $\sphericalangle BAC = 45^{\circ}$ and $ \sphericalangle ACB > 90^{\circ}.$
Show that $BC + (\sqrt{2} - 1)\cdot CA < AB.$
2 replies
pggp
Oct 22, 2020
parmenides51
Apr 8, 2021
J
Geometric inequality
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Reveal topic tags
geometry
geometric inequality
inequalities
L
G H BBookmark kLocked kLocked NReply
Source: Polish Junior Math Olympiad 2020 2nd Round
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pggp
89 posts
pggp
#1 h  Oct 22, 2020, 5:46 PM • 1 Y
Y by Amir Hossein
Let $ABC$ be such a triangle that $\sphericalangle BAC = 45^{\circ}$ and $ \sphericalangle ACB > 90^{\circ}.$
Show that $BC + (\sqrt{2} - 1)\cdot CA < AB.$
Z K Y
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cadaeibf
700 posts
cadaeibf
#2 h  Oct 23, 2020, 7:19 PM • 1 Y
Y by Amir Hossein
Let $D$ be the point on $AB$ such that $\angle ACD=\frac{\pi}{2}$. So $CD=CA$ and $AD=\sqrt{2}CA$. By triangle inequality on $BCD$, we have $BD+CD>BC$, so $AB=AD+BD=\sqrt{2}CA+BD>\sqrt{2}CA+BC-CD>BC+(\sqrt{2}-1)CA$
Z K Y
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parmenides51
30629 posts
parmenides51
#3 h  Apr 8, 2021, 5:26 PM
Y by
posted and solved also here
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