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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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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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true or false ?
SunnyEvan   1
N 21 minutes ago by SunnyEvan
Let $ a,b,c \geq 0 $ Prove that :
$$ \frac{3k}{k^4+k+1} \leq \frac{ka}{a+k^4b+kc}+\frac{kb}{b+k^4c+ka}+\frac{kc}{c+k^4a+kb} \leq k $$Where $ k \geq 1 $
1 reply
SunnyEvan
2 hours ago
SunnyEvan
21 minutes ago
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IMO Shortlist 2012, Combinatorics 1
lyukhson   75
N 21 minutes ago by damyan
Source: IMO Shortlist 2012, Combinatorics 1
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.

Proposed by Warut Suksompong, Thailand
75 replies
lyukhson
Jul 29, 2013
damyan
21 minutes ago
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Perfect Squares and a Prime Power
mojyla222   1
N 32 minutes ago by Quantum-Phantom
Source: IDMC 2025 P5
Find all natural numbers $a,b$ such that $a+1$ and $2(b+1)$ are both perfect squares and $a^2+b^2-1$ is a power of a prime number.


Proposed by Amirhossein Bateni
1 reply
mojyla222
Today at 5:07 AM
Quantum-Phantom
32 minutes ago
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Inspired by old results
sqing   6
N 35 minutes ago by SunnyEvan
Source: Own
Let $ a,b>0. $ Prove that
$$\frac{(a+1)^2}{b}+\frac{(b+k)^2}{a} \geq4(k+1) $$Where $ k\geq 0. $
$$\frac{a^2}{b}+\frac{(b+1)^2}{a} \geq4$$
6 replies
sqing
Yesterday at 2:43 AM
SunnyEvan
35 minutes ago
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Same radius geo
ThatApollo777   3
N 36 minutes ago by ThatApollo777
Source: Own
Classify all possible quadrupes of $4$ distinct points in a plane such the circumradius of any $3$ of them is the same.
3 replies
ThatApollo777
Yesterday at 7:37 AM
ThatApollo777
36 minutes ago
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Results on n students with distinct heights
Gloona   1
N an hour ago by quasar_lord
Source: CMI 2023 B4
In a class there are n students with unequal heights.
$\textbf{(a)}$ Find the number of orderings of the students such that the shortest person
is not at the front and the tallest person is not at the end.
$\textbf{(b)}$ Define the badness of an ordering as the maximum number $k$ such that there
are $k$ many people with height greater than in front of a person. For example:
the sequence $66, 61, 65, 64, 62, 70$ has badness $3$ since there are $3$ numbers greater
than $62$ in front of it. Let $f_k(n)$ denote the number of orderings of $n$ with badness $k$. Find $f_k(n)$.
hint
1 reply
Gloona
May 9, 2023
quasar_lord
an hour ago
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Iran second round 2025-q1
mohsen   2
N an hour ago by missionjoshi.65
Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.
2 replies
mohsen
Yesterday at 10:21 AM
missionjoshi.65
an hour ago
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From Recursion to Inequality
mojyla222   2
N 2 hours ago by missionjoshi.65
Source: IDMC 2025 P2
$\{a_n\}_{n\geq 1}$ is a sequence of real numbers with $a_1=1,\;a_2 =2$ such that for all $n\geq 1$
$$a_{n+2}=\dfrac{a_{n+1}^{2}}{1+a_{n}}+a_{n+1}.$$Prove that

$$\dfrac{1}{1+a_{1}+a_{2}}+\dfrac{1}{1+a_{2}+a_{3}}+\cdots + \dfrac{1}{1+a_{1403}+a_{1404}}>\dfrac{2^{1403}-1}{2^{1404}}.$$
Proposed by Mojtaba Zare
2 replies
mojyla222
Today at 5:01 AM
missionjoshi.65
2 hours ago
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Combinatorics
TUAN2k8   3
N 2 hours ago by TUAN2k8
A sequence of integers $a_1,a_2,...,a_k$ is call $k-balanced$ if it satisfies the following properties:
$i) a_i \neq a_j$ and $a_i+a_j \neq 0$ for all indices $i \neq j$.
$ii) \sum_{i=1}^{k} a_i=0$.
Find the smallest integer $k$ for which: Every $k-balanced$ sequence, there always exist two terms whose diffence is not less than $n$. (where $n$ is given positive integer)
3 replies
TUAN2k8
Yesterday at 8:22 AM
TUAN2k8
2 hours ago
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Inspired by Bet667
sqing   4
N 2 hours ago by SunnyEvan
Source: Own
Let $x,y\ge 0$ such that $k(x+y)=1+xy. $ Prove that $$x+y+\frac{1}{x}+\frac{1}{y}\geq 4k $$Where $k\geq 1. $
4 replies
sqing
Today at 2:34 AM
SunnyEvan
2 hours ago
J
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Very Easy Combinatorics Problem
zeta1   1
N 2 hours ago by expiredcraker
Ali and Veli goes to hunting. The probability that each will successfully hit a duck is $1/2$ on any given shot. During the hunt, Ali shoots $12$ times, and Veli shoots $13$ times. What is the probability that Veli hits more ducks than Ali?

$ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac{13}{25} \qquad\textbf{(C)}\ \dfrac{13}{24} \qquad\textbf{(D)}\ \dfrac{7}{13} \qquad\textbf{(E)}\ \dfrac{3}{4} $
1 reply
zeta1
3 hours ago
expiredcraker
2 hours ago
J
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Three variables inequality
Headhunter   2
N 3 hours ago by Headhunter
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
2 replies
Headhunter
5 hours ago
Headhunter
3 hours ago
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Inequalities
sqing   0
3 hours ago
Let $ x,y\geq 0 $ such that $ 2(x+y)=4+xy. $ Prove that$$x+y+\frac{1}{x}+\frac{1}{y}\geq5 $$Let $ x,y\geq 0 $ such that $ 3(x+y)=8+xy. $ Prove that$$x+y+\frac{1}{x}+\frac{1}{y}\geq5 $$Let $ x,y\geq 0 $ such that $ 3(x+y)=9+xy. $ Prove that$$x+y+\frac{1}{x}+\frac{1}{y}\geq \frac{20}{3} $$Let $ x,y\geq 0 $ such that $ 3(x+y)=6+xy. $ Prove that$$x+y+\frac{1}{x}+\frac{1}{y}\geq7-\frac{5}{\sqrt 3} $$
0 replies
sqing
3 hours ago
0 replies
J
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Inequalities
sqing   11
N 4 hours ago by sqing
Let $ a,b,c> 0 $ and $ \frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$. Prove that
$$ a+b+c\leq 3 $$
11 replies
sqing
Apr 4, 2025
sqing
4 hours ago
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J
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What is an isogonal conjugate and why is it useful?
EaZ_Shadow   6
N Apr 5, 2025 by maxamc
What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.
6 replies
EaZ_Shadow
Dec 28, 2024
maxamc
Apr 5, 2025
J
What is an isogonal conjugate and why is it useful?
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Reveal topic tags
geometry
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EaZ_Shadow
1228 posts
EaZ_Shadow
#1 h  Dec 28, 2024, 6:08 PM • 4 Y
Y by 1217662, 1218865, 1220118, Ad112358
What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.
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EaZ_Shadow
1228 posts
EaZ_Shadow
#2 h  Dec 29, 2024, 7:21 PM • 4 Y
Y by 1217662, 1218865, 1220118, Ad112358
/bump help please
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vanstraelen
8972 posts
vanstraelen
#3 h  Dec 29, 2024, 7:31 PM
Y by
https://en.wikipedia.org/wiki/Isogonal_conjugate
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EaZ_Shadow
1228 posts
EaZ_Shadow
#4 h  Dec 29, 2024, 7:41 PM • 4 Y
Y by 1217662, 1218865, 1220118, Ad112358
vanstraelen wrote:
https://en.wikipedia.org/wiki/Isogonal_conjugate

I don’t understand it though, can you explain it more simply
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greenturtle3141
3546 posts
greenturtle3141
#5 h  Dec 29, 2024, 9:00 PM
Y by
Can you provide an example of the appearance of the isogonal conjugate in Olympiad geometry? This can help explain why it was needed for that particular problem.
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EaZ_Shadow
1228 posts
EaZ_Shadow
#6 h  Dec 29, 2024, 9:47 PM • 4 Y
Y by 1217662, 1218865, 1220118, Ad112358
greenturtle3141 wrote:
Can you provide an example of the appearance of the isogonal conjugate in Olympiad geometry? This can help explain why it was needed for that particular problem.

Like in general
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maxamc
548 posts
maxamc
#7 h  Apr 5, 2025, 2:40 PM
Y by
it mean you have a point p. draw line from ap bp and cp. reflec the line about angle bisector for 3 new liine that intersec at p'. p abd p' are isogonal conjucates. for example orthoscenter and circumsceter are isogbal conjuacates.
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