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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 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
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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primes,exponentials,factorials
skellyrah   0
5 minutes ago
find all primes p,q such that $$ \frac{p^q-q^p-p-q}{p!-q!} $$is a prime number
0 replies
skellyrah
5 minutes ago
0 replies
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4 variables with quadrilateral sides 2
mihaig   1
N 7 minutes ago by mihaig
Source: Own
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$$
1 reply
mihaig
Yesterday at 8:47 PM
mihaig
7 minutes ago
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Good divisors and special numbers.
Nuran2010   3
N 8 minutes ago by Nuran2010
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
$N$ is a positive integer. Call all positive divisors of $N$ which are different from $1$ and $N$ beautiful divisors.We call $N$ a special number when it has at least $2$ beautiful divisors and difference of any $2$ beautiful divisors divides $N$ as well. Find all special numbers.
3 replies
Nuran2010
Yesterday at 4:52 PM
Nuran2010
8 minutes ago
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Functionnal equation
Rayanelba   1
N 9 minutes ago by Mrcuberoot
Source: Own
Find all functions $f:\mathbb{R}_{>0}\to \mathbb{R}_{>0}$ that verify the following equation for all $x,y\in \mathbb{R}_{>0}$:
$f(x+yf(f(x)))+f(\frac{x}{y})=\frac{x}{y}+f(x+xy)$
1 reply
Rayanelba
2 hours ago
Mrcuberoot
9 minutes ago
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An I for an I
Eyed   66
N 9 minutes ago by SimplisticFormulas
Source: 2020 ISL G8
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.

Show that $A,X,Y$ are collinear.
66 replies
Eyed
Jul 20, 2021
SimplisticFormulas
9 minutes ago
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Find f
Redriver   5
N 10 minutes ago by cefer
Find all $: R \to R : \ \ f(x^2+f(y))=y+f^2(x)$
5 replies
Redriver
Jun 25, 2006
cefer
10 minutes ago
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Geometric inequality with Fermat point
Assassino9931   6
N 27 minutes ago by arqady
Source: Balkan MO Shortlist 2024 G2
Let $ABC$ be an acute triangle and let $P$ be an interior point for it such that $\angle APB = \angle BPC = \angle CPA$. Prove that
$$ \frac{PA^2 + PB^2 + PC^2}{2S} + \frac{4}{\sqrt{3}} \leq \frac{1}{\sin \alpha} + \frac{1}{\sin \beta} + \frac{1}{\sin \gamma}. $$When does equality hold?
6 replies
Assassino9931
Apr 27, 2025
arqady
27 minutes ago
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Straight-edge is a social construct
anantmudgal09   27
N 30 minutes ago by cj13609517288
Source: INMO 2023 P6
Euclid has a tool called cyclos which allows him to do the following:
[list]
[*] Given three non-collinear marked points, draw the circle passing through them.
[*] Given two marked points, draw the circle with them as endpoints of a diameter.
[*] Mark any intersection points of two drawn circles or mark a new point on a drawn circle.
[/list]

Show that given two marked points, Euclid can draw a circle centered at one of them and passing through the other, using only the cyclos.

Proposed by Rohan Goyal, Anant Mudgal, and Daniel Hu
27 replies
anantmudgal09
Jan 15, 2023
cj13609517288
30 minutes ago
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Reducibility of 2x^2 cyclotomic
vincentwant   2
N 42 minutes ago by vincentwant
Let $S$ denote the set of all positive integers less than $1020$ that are relatively prime to $1020$. Let $\omega=\cos\frac{\pi}{510}+i\sin\frac{\pi}{510}$. Is the polynomial $$\prod_{n\in S}(2x^2-\omega^n)$$reducible over the rational numbers, given that it has integer coefficients?
2 replies
vincentwant
3 hours ago
vincentwant
42 minutes ago
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Weighted Blocks
ilovemath04   51
N an hour ago by Maximilian113
Source: ISL 2019 C2
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.
51 replies
ilovemath04
Sep 22, 2020
Maximilian113
an hour ago
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Very easy NT
GreekIdiot   7
N an hour ago by Primeniyazidayi
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
7 replies
GreekIdiot
4 hours ago
Primeniyazidayi
an hour ago
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Azer and Babek playing a game on a chessboard
Nuran2010   1
N an hour ago by Diamond-jumper76
Source: Azerbaijan Al-Khwarizmi IJMO TST
Azer and Babek have a $8 \times 8$ chessboard. Initially, Azer colors all cells of this chessboard with some colors. Then, Babek takes $2$ rows and $2$ columns and looks at the $4$ cells in the intersection. Babek wants to have all these $4$ cells in a same color, but Azer doesn't. With at least how many colors, Azer can reach his goal?
1 reply
Nuran2010
Yesterday at 5:03 PM
Diamond-jumper76
an hour ago
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Something nice
KhuongTrang   27
N an hour ago by arqady
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
27 replies
1 viewing
KhuongTrang
Nov 1, 2023
arqady
an hour ago
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Hard inequality
JK1603JK   4
N an hour ago by JK1603JK
Source: unknown?
Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $$P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$
4 replies
JK1603JK
Yesterday at 4:24 AM
JK1603JK
an hour ago
J
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J
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Construct a triangle with area ≤ A (IMO SL 1987-P19)
Amir Hossein   3
N Aug 18, 2016 by lebathanh
Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than
\[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\]

Proposed by Soviet Union
3 replies
Amir Hossein
Aug 19, 2010
lebathanh
Aug 18, 2016
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Construct a triangle with area ≤ A (IMO SL 1987-P19)
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Reveal topic tags
geometry
trigonometry
geometric inequality
Trigonometric inequality
IMO Shortlist
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Amir Hossein
5452 posts
Amir Hossein
#1 h  Aug 19, 2010, 10:57 AM • 2 Y
Y by Adventure10, Mango247
Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than
\[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\]

Proposed by Soviet Union
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Learner94
634 posts
Learner94
#2 h  Nov 9, 2011, 5:47 AM • 2 Y
Y by Adventure10, Mango247
I will denote $\alpha ,\beta, \gamma$ by $a,b,c$ .W.L.O.G $a\geq b\geq c$. Since $b+c>a$, we have $a< \frac{\pi}{2}$.For existence of a triangle with side lengths as $\sin a, \sin b , \sin c$ its necessary and sufficient to have $\sin b + \sin c > \sin a$.
If $b+c<\frac{\pi}{2}$, then $\sin b + \sin c >\sin(b+c) > \sin a$, If $b+c>\frac{\pi}{2}$, then $a< \pi - b-c <\frac{\pi}{2}$ so again $\sin b + \sin c >\sin(b+c) > \sin a$.
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sayantanchakraborty
505 posts
sayantanchakraborty
#3 h  Jun 2, 2014, 6:51 AM • 2 Y
Y by Adventure10, Mango247
My proof of the first part is the same as Learner 94,and it may essentially the same for all.However can someone give a proof of the second part.Or,I think I am missing something,maybe its too easy.
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lebathanh
464 posts
lebathanh
#4 h  Aug 18, 2016, 6:33 AM • 2 Y
Y by Adventure10, Mango247
why don't we dicuss it i think it is very interesting
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