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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 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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Inequalities
sqing   4
N 16 minutes ago by alexheinis
Let $a,b,c >1 $ and $ \frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=1.$ Show that$$ab+bc+ca \geq 48$$$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq \frac{3}{4}$$Let $a,b,c >1 $ and $ \frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=2.$ Show that$$ab+bc+ca \geq \frac{75}{4}$$$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq \frac{6}{5}$$Let $a,b,c >1 $ and $ \frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=3.$ Show that$$ab+bc+ca \geq 12$$$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq \frac{3}{2}$$
4 replies
sqing
2 hours ago
alexheinis
16 minutes ago
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Find function
trito11   1
N an hour ago by Mathzeus1024
Find $f:\mathbb{R^+} \to \mathbb{R^+} $ such that
i) f(x)>f(y) $\forall$ x>y>0
ii) f(2x)$\ge$2f(x)$\forall$x>0
iii)$f(f(x)f(y)+x)=f(xf(y))+f(x)$$\forall$x,y>0
1 reply
trito11
Nov 11, 2019
Mathzeus1024
an hour ago
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The centroid of ABC lies on ME [2023 Abel, Problem 1b]
Amir Hossein   2
N 2 hours ago by MITDragon
In the triangle $ABC$, points $D$ and $E$ lie on the side $BC$, with $CE = BD$. Also, $M$ is the midpoint of $AD$. Show that the centroid of $ABC$ lies on $ME$.
2 replies
Amir Hossein
Mar 12, 2024
MITDragon
2 hours ago
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min A=x+1/x+y+1/y if 2(x+y)=1+xy for x,y>0 , 2020 ISL A3 for juniors
parmenides51   13
N 2 hours ago by AylyGayypow009
Source: 2021 Greece JMO p1 (serves also as JBMO TST) / based on 2020 IMO ISL A3
If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$
13 replies
parmenides51
Jul 21, 2021
AylyGayypow009
2 hours ago
J
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Another geo P1
alchemyst_   32
N 2 hours ago by tilya_TASh
Source: Balkan MO 2022 P1
Let $ABC$ be an acute triangle such that $CA \neq CB$ with circumcircle $\omega$ and circumcentre $O$. Let $t_A$ and $t_B$ be the tangents to $\omega$ at $A$ and $B$ respectively, which meet at $X$. Let $Y$ be the foot of the perpendicular from $O$ onto the line segment $CX$. The line through $C$ parallel to line $AB$ meets $t_A$ at $Z$. Prove that the line $YZ$ passes through the midpoint of the line segment $AC$.

Proposed by Dominic Yeo, United Kingdom
32 replies
+1 w
alchemyst_
May 6, 2022
tilya_TASh
2 hours ago
J
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Set Partition
Butterfly   1
N 2 hours ago by aaravdodhia
For the set of positive integers $\{1,2,…,n\}(n\ge 3)$, no matter how its elements are partitioned into two subsets, at least one of the subsets must contain three numbers $a,b,c$ ($a=b$ is allowed) such that $ab=c$. Find the minimal $n$.
1 reply
Butterfly
Yesterday at 1:06 AM
aaravdodhia
2 hours ago
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Thailand MO 2025 P2
Kaimiaku   1
N 3 hours ago by totalmathguy
A school sent students to compete in an academic olympiad in $11$ differents subjects, each consist of $5$ students. Given that for any $2$ different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least $4$ different subjects.
1 reply
Kaimiaku
3 hours ago
totalmathguy
3 hours ago
J
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Sum and product of digits
Sadigly   5
N 3 hours ago by Bergo1305
Source: Azerbaijan NMO 2018
For a positive integer $n$, define $f(n)=n+P(n)$ and $g(n)=n\cdot S(n)$, where $P(n)$ and $S(n)$ denote the product and sum of the digits of $n$, respectively. Find all solutions to $f(n)=g(n)$
5 replies
Sadigly
Sunday at 9:19 PM
Bergo1305
3 hours ago
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Anything real in this system must be integer
Assassino9931   4
N 3 hours ago by Leman_Nabiyeva
Source: Al-Khwarizmi International Junior Olympiad 2025 P1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

Marek Maruin, Slovakia
4 replies
Assassino9931
May 9, 2025
Leman_Nabiyeva
3 hours ago
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Oh my god
EeEeRUT   1
N 3 hours ago by ItzsleepyXD
Source: TMO 2025 P5
In a class, there are $n \geqslant 3$ students and a teacher with $M$ marbles. The teacher then play a Marble distribution according to the following rules. At the start, the teacher distributed all her marbles to students, so that each student receives at least $1$ marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of $2(n-1)$. That chosen student then equally distribute half of his marbles to $n-1$ other students. The same goes on until the teacher is not able to choose anymore student.

Find all integer $M$, such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.
1 reply
EeEeRUT
4 hours ago
ItzsleepyXD
3 hours ago
J
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Inequalities
sqing   6
N 3 hours ago by sqing
Let $ a,b,c\geq 0 , (a+8)(b+c)=9.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{38}{23}$$Let $ a,b,c\geq 0 , (a+2)(b+c)=3.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{2(2\sqrt{3}+1)}{5}$$
6 replies
sqing
May 10, 2025
sqing
3 hours ago
J
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Find all integers satisfying this equation
Sadigly   1
N 3 hours ago by aaravdodhia
Source: Azerbaijan NMO 2019
Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$
1 reply
Sadigly
Sunday at 8:30 PM
aaravdodhia
3 hours ago
J
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A geometry problem involving 2 circles
Ujiandsd   1
N 3 hours ago by Ujiandsd
Source: L
Point M is the midpoint of side BC of triangle ABC. The length of the radius of the outer circle of triangle ABM, triangle ACM
is 5 and 7 respectively find the distance between the center of their outer circles
1 reply
Ujiandsd
May 11, 2025
Ujiandsd
3 hours ago
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Inspired by lbh_qys.
sqing   4
N 3 hours ago by sqing
Source: Own
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +b+1}+ \frac{b}{b^2+a +b+1} \leq \frac{1}{2} $$$$ \frac{a}{a^2+ab+a+b+1}+ \frac{b}{b^2+ab+a+b+1} \leq \sqrt 2-1 $$$$\frac{a}{a^2+ab+a+1}+ \frac{b}{b^2+ab+b+1} \leq \frac{2(2\sqrt 2-1)}{7} $$$$\frac{a}{a^2+ab+b+1}+ \frac{b}{b^2+ab+a+1} \leq \frac{2(2\sqrt 2-1)}{7} $$
4 replies
sqing
Today at 3:45 AM
sqing
3 hours ago
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J
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Three variables inequality
Headhunter   6
N Apr 30, 2025 by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
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.
6 replies
Headhunter
Apr 20, 2025
lbh_qys
Apr 30, 2025
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Three variables inequality
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Reveal topic tags
inequalities
algebra
proof-writing
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Headhunter
1963 posts
Headhunter
#1 h  Apr 20, 2025, 6:58 AM
Y by
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
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.
Z K Y
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programjames1
3046 posts
programjames1
#2 h  Apr 20, 2025, 7:20 AM
Y by
Can you give the reference? I think this is from a USAJMO contest in the 2010s.

EDIT: I was thinking of USA(J)MO 2018 #1 (#2) which can be rearranged to a similar inequality.
This post has been edited 2 times. Last edited by programjames1, Apr 20, 2025, 7:23 AM
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Headhunter
1963 posts
Headhunter
#3 h  Apr 20, 2025, 9:01 AM
Y by
I guess that this problem is from chinese materials at 1990~2004. but I'm not sure. Thanks.
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lbh_qys
574 posts
lbh_qys
#4 h  Apr 21, 2025, 3:04 AM
Y by
Hint
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lbh_qys
574 posts
lbh_qys
#5 h  Apr 21, 2025, 3:18 AM • 2 Y
Y by programjames1, spy27
another solution
This post has been edited 1 time. Last edited by lbh_qys, Apr 21, 2025, 3:18 AM
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spy27
8 posts
spy27
#6 h  Apr 30, 2025, 5:38 AM
Y by
lbh_qys wrote:
another solution

Can you explain the \( a + b + c \neq 0 \) case in some detail?
Z K Y
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lbh_qys
574 posts
lbh_qys
#7 h  Apr 30, 2025, 6:08 AM
Y by
spy27 wrote:
lbh_qys wrote:
another solution

Can you explain the \( a + b + c \neq 0 \) case in some detail?

$f(x) = (a+x)^2 + (b+x)^2 + (c+x)^2 $ get minimum at $x=0$ iff $a+b+c=0$.
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