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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
4 hours ago
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
4 hours ago
0 replies
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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 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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A property of divisors
rightways   10
N an hour ago by akliu
Source: Kazakhstan NMO 2016, P1
Prove that one can arrange all positive divisors of any given positive integer around a circle so that for any two neighboring numbers one is divisible by another.
10 replies
rightways
Mar 17, 2016
akliu
an hour ago
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Famous geo configuration appears on the district MO
AndreiVila   3
N an hour ago by chirita.andrei
Source: Romanian District Olympiad 2025 10.4
Let $ABCDEF$ be a convex hexagon with $\angle A = \angle C=\angle E$ and $\angle B = \angle D=\angle F$.
[list=a]
[*] Prove that there is a unique point $P$ which is equidistant from sides $AB,CD$ and $EF$.
[*] If $G_1$ and $G_2$ are the centers of mass of $\triangle ACE$ and $\triangle BDF$, show that $\angle G_1PG_2=60^{\circ}$.
3 replies
AndreiVila
Mar 8, 2025
chirita.andrei
an hour ago
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linear algebra
ay19bme   5
N an hour ago by loup blanc
Does the matrix equation $X^3=mI_2$ is solvable over $M_{2}(\mathbb{Z})$ for every $m\in \mathbb{Z}$. Here $X\in M_{2}(\mathbb{Z})$, $I_2=\begin{pmatrix} 1& 0\\0 & 1\end{pmatrix}$.
5 replies
ay19bme
Today at 7:06 AM
loup blanc
an hour ago
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kind of well known?
dotscom26   2
N an hour ago by alexheinis
Source: MBL
Let $ y_1, y_2, ..., y_{2025}$ be real numbers satisfying
$ y_1^2 + y_2^2 + \cdots + y_{2025}^2 = 1. $
Find the maximum value of
$ |y_1 - y_2| + |y_2 - y_3| + \cdots + |y_{2025} - y_1|. $

I have seen many problems with the same structure, Id really appreciate if someone could explain which approach is suitable here
2 replies
dotscom26
Yesterday at 4:11 AM
alexheinis
an hour ago
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hard problem
Cobedangiu   0
an hour ago
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
0 replies
Cobedangiu
an hour ago
0 replies
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Ornaments and Christmas trees
Morskow   30
N an hour ago by akliu
Source: Slovenia IMO TST 2018, Day 1, Problem 1
Let $n$ be a positive integer. On the table, we have $n^2$ ornaments in $n$ different colours, not necessarily $n$ of each colour. Prove that we can hang the ornaments on $n$ Christmas trees in such a way that there are exactly $n$ ornaments on each tree and the ornaments on every tree are of at most $2$ different colours.
30 replies
Morskow
Dec 17, 2017
akliu
an hour ago
J
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Informatics Competition (OTIS MOCK AIME 2025 II #11)
YaoAOPS   2
N an hour ago by akliu
Source: OTIS MOCK AIME 2025 II
At an informatics competition each student earns a score in $\{0, 1, \dots, 100\}$ on each of six problems, and their total score is the sum of the six scores (out of $600$). Given two students $A$ and $B$, we write $A \succ B$ if there are at least five problems on which $A$ scored strictly higher than $B$.

Compute the smallest integer $c$ such that the following statement is true: for every integer $n \ge 2$, given students $A_1$, \dots, $A_n$ satisfying $A_1 \succ A_2 \succ \dots \succ A_n$, the total score of $A_n$ is always at most $c$ points more than the total score of $A_1$.

Jiahe Liu
2 replies
YaoAOPS
Jan 22, 2025
akliu
an hour ago
J
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Polynomial meets geometry
chirita.andrei   0
2 hours ago
Source: Own. Proposed for Romanian National Olympiad 2025.
(a) Let $A,B,C$ be collinear points (in order) and $D$ a point in plane. Consider the disc $\mathcal{D}$ of center $D$ and radius $kBD$, for some $k\in(0,1)$. Prove that $\mathcal{D}\cap [AC]$ is either the empty set or a segment of length at most $2kAC$.
(b) Let $n$ be a positive integer and $P(X)\in\mathbb{C}[X]$ be a polynomial of degree $n$. Prove that \[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]
0 replies
chirita.andrei
2 hours ago
0 replies
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On units in a ring with a polynomial property
Ciobi_   2
N 2 hours ago by Toinfinity
Source: Romania NMO 2025 12.1
We say a ring $(A,+,\cdot)$ has property $(P)$ if :
\[ \begin{cases} \text{the set } A \text{ has at least } 4 \text{ elements} \\ \text{the element } 1+1 \text{ is invertible}\\ x+x^4=x^2+x^3 \text{ holds for all } x \in A \end{cases} \]a) Prove that if a ring $(A,+,\cdot)$ has property $(P)$, and $a,b \in A$ are distinct elements, such that $a$ and $a+b$ are units, then $1+ab$ is also a unit, but $b$ is not a unit.
b) Provide an example of a ring with property $(P)$.
2 replies
Ciobi_
5 hours ago
Toinfinity
2 hours ago
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Thanks u!
Ruji2018252   1
N 2 hours ago by alexheinis
Let $a_1,...,a_{2024}\in\mathbb{Z}$ and $1\leqslant a_1\leqslant a_2\leqslant ...\leqslant a_{2024}\leqslant 99$ and
\[P=a_1^2+a_2^2+...+a_{2024}^2-(a_1a_3+a_2a_4+...+a_{2022}a_{2024})\]Find maximum $P$
1 reply
Ruji2018252
Yesterday at 11:51 AM
alexheinis
2 hours ago
J
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Number-erasing game with multiples of 3
p108771953   4
N 2 hours ago by conejita
Source: 2024 Mexican Mathematical Olympiad, Problem 6
Ana and Beto play on a blackboard where all integers from 1 to 2024 (inclusive) are written. On each turn, Ana chooses three numbers $a,b,c$ written on the board and then Beto erases them and writes one of the following numbers:
$$a+b-c, a-b+c, ~\text{or}~ -a+b+c.$$The game ends when only two numbers are left on the board and Ana cannot play. If the sum of the final numbers is a multiple of 3, Beto wins. Otherwise, Ana wins. ¿Who has a winning strategy?
4 replies
p108771953
Nov 6, 2024
conejita
2 hours ago
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Another square grid :D
MathLuis   43
N 2 hours ago by akliu
Source: USEMO 2021 P1
Let $n$ be a fixed positive integer and consider an $n\times n$ grid of real numbers. Determine the greatest possible number of cells $c$ in the grid such that the entry in $c$ is both strictly greater than the average of $c$'s column and strictly less than the average of $c$'s row.

Proposed by Holden Mui
43 replies
MathLuis
Oct 30, 2021
akliu
2 hours ago
J
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Number theory
Maaaaaaath   0
2 hours ago
Let $m$ be a positive integer . Prove that there exists infinitely many pairs of positive integers $(x,y)$ such that $\gcd(x,y)=1$ and :

$$xy | x^2+y^2+m$$
0 replies
Maaaaaaath
2 hours ago
0 replies
J
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Integral inequality with differentiable function
Ciobi_   1
N 4 hours ago by MS_asdfgzxcvb
Source: Romania NMO 2025 12.2
Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]
1 reply
Ciobi_
5 hours ago
MS_asdfgzxcvb
4 hours ago
J
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J
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Derivative of function R^2 to R^2
Sifan.C.Maths   1
N Mar 22, 2025 by alexheinis
Source: Internet
Give a function $f:\mathbb{R}^2 \to \mathbb{R}^2: f(x,y)=(x^2+xy,y^2+x)$. Calculate the first and second derivative of the function at the point $(1,-1)$.
1 reply
Sifan.C.Maths
Mar 22, 2025
alexheinis
Mar 22, 2025
J
Derivative of function R^2 to R^2
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Reveal topic tags
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Sifan.C.Maths
41 posts
Sifan.C.Maths
#1 h  Mar 22, 2025, 7:09 AM
Y by
Give a function $f:\mathbb{R}^2 \to \mathbb{R}^2: f(x,y)=(x^2+xy,y^2+x)$. Calculate the first and second derivative of the function at the point $(1,-1)$.
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alexheinis
10515 posts
alexheinis
#2 h  Mar 22, 2025, 3:38 PM • 1 Y
Y by Sifan.C.Maths
In the point $(x,y)$ we have $Df\in L(R^2,R^2)$ and $Df=\begin{pmatrix} 2x+y& x\\1&2y\end{pmatrix}$. To simplify notation we identify $M_2(R)$ with $R^4$ and then $Df=(2x+y,x,1,2y)$. Then $D^2f=\begin{pmatrix} 2&1\\1&0\\0&0\\0&2\end{pmatrix}$. To be exact, it is the linear map with $(1,0)\rightarrow \begin{pmatrix} 2&1\\0&0\end{pmatrix}$ and $(0,1)\rightarrow \begin{pmatrix} 1&0\\0&2\end{pmatrix}$.
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