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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
5 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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[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers[/list]
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0 replies
jlacosta
5 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
J
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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   5
N 15 minutes ago by Primeniyazidayi
problem
5 replies
Cobedangiu
Mar 31, 2025
Primeniyazidayi
15 minutes ago
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Geo Final but hard to solve with Conics...
Seungjun_Lee   5
N 16 minutes ago by L13832
Source: 2025 Korea Winter Program Practice Test P4
Let $\omega$ be the circumcircle of triangle $ABC$ with center $O$, and the $A$ inmixtilinear circle is tangent to $AB, AC, \omega$ at $D,E,T$ respectively. $P$ is the intersection of $TO$ and $DE$ and $X$ is the intersection of $AP$ and $\omega$. Prove that the isogonal conjugate of $P$ lies on the line passing through the midpoint of $BC$ and $X$.
5 replies
Seungjun_Lee
Jan 18, 2025
L13832
16 minutes ago
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Thanks u!
Ruji2018252   3
N 17 minutes ago by Primeniyazidayi
Let $x,y,z,t\in\mathbb{R}$ and $\begin{cases}x^2+y^2=4\\z^2+t^2=9\\xt+yz\geqslant 6\end{cases}$.
$1,$ Prove $xz=yt$
$2,$ Find maximum $P=x+z$
3 replies
Ruji2018252
Mar 30, 2025
Primeniyazidayi
17 minutes ago
J
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Vieta's Polynomial x^20-7x^3+1=0
Goblik   3
N 25 minutes ago by cazanova19921
If $x_1,x_2,...,x_{20}$ are roots of $x^{20}-7x^3+1=0$, then find $\frac{1}{x_1^{2}+1}+\frac{1}{x_2^{2}+1}+...+\frac{1}{x_{20}^{2}+1}$
3 replies
Goblik
an hour ago
cazanova19921
25 minutes ago
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Ways to Place Counters on 2mx2n board
EpicParadox   37
N an hour ago by akliu
Source: 2019 Canadian Mathematical Olympiad Problem 3
You have a $2m$ by $2n$ grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place $mn$ counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
37 replies
EpicParadox
Mar 28, 2019
akliu
an hour ago
J
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Number theory
Maaaaaaath   1
N an hour ago by CHESSR1DER
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 :

$$xy | x^2+y^2+m$$
1 reply
Maaaaaaath
4 hours ago
CHESSR1DER
an hour ago
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Problem 4 from IMO 1997
iandrei   28
N an hour ago by akliu
Source: IMO Shortlist 1997, Q4
An $ n \times n$ matrix whose entries come from the set $ S = \{1, 2, \ldots , 2n - 1\}$ is called a silver matrix if, for each $ i = 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that:

(a) there is no silver matrix for $ n = 1997$;

(b) silver matrices exist for infinitely many values of $ n$.
28 replies
iandrei
Jul 28, 2003
akliu
an hour ago
J
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2025 Caucasus MO Seniors P8
BR1F1SZ   1
N an hour ago by sami1618
Source: Caucasus MO
Determine for which integers $n \geqslant 4$ the cells of a $1 \times (2n+1)$ table can be filled with the numbers $1, 2, 3, \dots, 2n + 1$ such that the following conditions are satisfied:
[list=i]
[*]Each of the numbers $1, 2, 3, \dots, 2n + 1$ appears exactly once.
[*]In any $1 \times 3$ rectangle, one of the numbers is the arithmetic mean of the other two.
[*]The number $1$ is located in the middle cell of the table.
[/list]
1 reply
BR1F1SZ
Mar 26, 2025
sami1618
an hour ago
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Unlimited candy in PAGMO
JuanDelPan   21
N 2 hours ago by akliu
Source: Pan-American Girls' Mathematical Olympiad 2021, P5
Celeste has an unlimited amount of each type of $n$ types of candy, numerated type 1, type 2, ... type n. Initially she takes $m>0$ candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:

$1.$ She eats a candy of type $k$, and in its position in the row she places one candy type $k-1$ followed by one candy type $k+1$ (we consider type $n+1$ to be type 1, and type 0 to be type $n$).

$2.$ She chooses two consecutive candies which are the same type, and eats them.

Find all positive integers $n$ for which Celeste can leave the table empty for any value of $m$ and any configuration of candies on the table.

$\textit{Proposed by Federico Bach and Santiago Rodriguez, Colombia}$
21 replies
JuanDelPan
Oct 6, 2021
akliu
2 hours ago
J
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set with c+2a>3b
VicKmath7   48
N 2 hours ago by akliu
Source: ISL 2021 A1
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.

Proposed by Dominik Burek and Tomasz Ciesla, Poland
48 replies
VicKmath7
Jul 12, 2022
akliu
2 hours ago
J
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A property of divisors
rightways   10
N 2 hours 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
2 hours ago
J
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Famous geo configuration appears on the district MO
AndreiVila   3
N 3 hours 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
3 hours ago
J
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kind of well known?
dotscom26   2
N 3 hours 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
3 hours ago
J
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hard problem
Cobedangiu   0
3 hours 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
3 hours ago
0 replies
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J
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Easy algebra
Namisgood   1
N Mar 31, 2025 by ND_
Source: Indian mathematics olympiad Stage-I 2014(PRMO)
For a natural number b, let N(b) denote the number of natural numbers for which the equation x^2+ax+b has integer roots. What is the smallest value of b for which N(b)=20
1 reply
Namisgood
Mar 31, 2025
ND_
Mar 31, 2025
J
Easy algebra
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Reveal topic tags
algebra
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Source: Indian mathematics olympiad Stage-I 2014(PRMO)
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Namisgood
1 post
Namisgood
#1 h  Mar 31, 2025, 10:39 AM
Y by
For a natural number b, let N(b) denote the number of natural numbers for which the equation x^2+ax+b has integer roots. What is the smallest value of b for which N(b)=20
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ND_
32 posts
ND_
#2 h  Mar 31, 2025, 10:45 AM • 1 Y
Y by Namisgood
https://artofproblemsolving.com/community/c918954_2014_india_prmo
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