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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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Find all functions $f$ is strictly increasing : \(\mathbb{R^+}\) \(\rightarrow\)
guramuta   3
N 18 minutes ago by ja.
Find all functions $f$ is strictly increasing : \(\mathbb{R^+}\) \(\rightarrow\) \(\mathbb{R^+}\) such that:
i) $f(2x)$ \(\geq\) $2f(x)$
ii) $f(f(x)f(y)+x) = f(xf(y)) + f(x) $
3 replies
guramuta
Yesterday at 1:45 PM
ja.
18 minutes ago
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Fourth powers and square roots
willwin4sure   39
N 24 minutes ago by awesomeming327.
Source: USA TSTST 2020 Problem 4, by Yang Liu
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]

Yang Liu
39 replies
willwin4sure
Dec 14, 2020
awesomeming327.
24 minutes ago
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Interesting inequalities
sqing   1
N 27 minutes ago by sqing
Source: Own
Let $ a,b >0 $ and $ a^2-ab+b^2\leq 1 $ . Prove that
$$a^4 +b^4+\frac{a }{b +1}+ \frac{b }{a +1} \leq 3$$$$a^3 +b^3+\frac{a^2}{b^2+1}+ \frac{b^2}{a^2+1} \leq 3$$$$a^4 +b^4-\frac{a}{b+1}-\frac{b}{a+1} \leq 1$$$$a^4+b^4 -\frac{a^2}{b^2+1}- \frac{b^2}{a^2+1}\leq 1$$$$a^3+b^3 -\frac{a^3}{b^3+1}- \frac{b^3}{a^3+1}\leq 1$$
1 reply
1 viewing
sqing
30 minutes ago
sqing
27 minutes ago
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Sum of 1/(a^5(b+2c))^2 at least 1/3 [USA TST 2010 2]
MellowMelon   42
N 33 minutes ago by Adywastaken
Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]
42 replies
MellowMelon
Jul 26, 2010
Adywastaken
33 minutes ago
J
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Weird function?
ItzsleepyXD   2
N an hour ago by ItzsleepyXD
Source: Own
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \),
\[ f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y). \]
2 replies
ItzsleepyXD
Apr 11, 2025
ItzsleepyXD
an hour ago
J
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Almost similar one but more answer lol
ItzsleepyXD   0
an hour ago
Source: Own , Modified
Find all non decreasing functions or non increasing function $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$

$$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(f(x)))+y$$.
0 replies
ItzsleepyXD
an hour ago
0 replies
J
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A lot of unexpected answer from non decreasing function
ItzsleepyXD   0
an hour ago
Source: Own
Find all non decreasing function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ and $m,n \in \mathbb{N}_0$ such that $m+n \neq 0$ there exist $m',n' \in \mathbb{N}_0$ such that $m'+n'=m+n+1$ and $$f(f^m(x)+f^n(y))=f^{m'}(x)+f^{n'}(y)$$. Note : $f^0(x)=x$ and $f^{n}(x)=f(f^{n-1}(x))$ for all $n \in \mathbb{N}$ . original
0 replies
ItzsleepyXD
an hour ago
0 replies
J
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Cute Inequality
EthanWYX2009   0
2 hours ago
Let $a_1,\ldots ,a_n\in\mathbb R\backslash\{0\},$ determine the minimum and maximum value of
\[\frac{\sum_{i,j=1}^n|a_i+a_j|}{\sum_{i=1}^n|a_i|}.\]
0 replies
EthanWYX2009
2 hours ago
0 replies
J
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Interesting inequality
sealight2107   3
N 2 hours ago by NguyenVanHoa29
Source: Own
Let $a,b,c>0$ such that $a+b+c=3$. Find the minimum value of:
$Q=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{1}{a^3+b^3+abc}+\frac{1}{b^3+c^3+abc}+\frac{1}{c^3+a^3+abc}$
3 replies
sealight2107
May 6, 2025
NguyenVanHoa29
2 hours ago
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Is this FE is solvable?
ItzsleepyXD   0
2 hours ago
Source: Own , If not appear somewhere before
Find all function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ . $$f(x+f(y))+f(x+y)=2x+f(y)+f(f(y))$$. Original
0 replies
ItzsleepyXD
2 hours ago
0 replies
J
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Symmetry in Circumcircle Intersection
Mimii08   2
N 2 hours ago by mashumaro
Hi! Here's another geometry problem I'm thinking about, and I would appreciate any help with a proof. Thanks in advance!

Let AD and BE be the altitudes of an acute triangle ABC, with D on BC and E on AC. The line DE intersects the circumcircle of triangle ABC again at two points M and N. Prove that CM = CN.

Thanks for your time and help!
2 replies
Mimii08
4 hours ago
mashumaro
2 hours ago
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Blue chessboard
rcorreaa   10
N 2 hours ago by Jaxman8
Source: 2022 Brazilian National Mathematical Olympiad - Problem 6
Some cells of a $10 \times 10$ are colored blue. A set of six cells is called gremista when the cells are the intersection of three rows and two columns, or two rows and three columns, and are painted blue. Determine the greatest value of $n$ for which it is possible to color $n$ chessboard cells blue such that there is not a gremista set.
10 replies
rcorreaa
Nov 22, 2022
Jaxman8
2 hours ago
J
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Substitution
JCE   3
N 2 hours ago by K124659
I've been working on this for about an hour or so, and I can't get this problem. I know the answer, but no idea on how to find it.
Please help?

2x-y^2=4
x^2+y=14
3 replies
JCE
May 27, 2006
K124659
2 hours ago
J
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Something nice
KhuongTrang   34
N 3 hours ago by TNKT
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
34 replies
KhuongTrang
Nov 1, 2023
TNKT
3 hours ago
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J
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Minimal number of divisors
Garfield   6
N Apr 12, 2025 by Namisgood
Source: Serbian junior TST 2nd
Find minimal number of divisors that can number $|2016^m-36^n|$ have,where $m$ and $n$ are natural numbers.
6 replies
Garfield
May 21, 2016
Namisgood
Apr 12, 2025
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Minimal number of divisors
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Reveal topic tags
number theory
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Source: Serbian junior TST 2nd
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Garfield
243 posts
Garfield
#1 h  May 21, 2016, 7:15 PM • 3 Y
Y by Adventure10, Mango247, AlexCenteno2007
Find minimal number of divisors that can number $|2016^m-36^n|$ have,where $m$ and $n$ are natural numbers.
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tenplusten
1000 posts
tenplusten
#2 h  May 21, 2016, 7:21 PM • 2 Y
Y by Adventure10, Mango247
Hi dude where can I find other Serbian Junior TST problems?
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Garfield
243 posts
Garfield
#3 h  May 21, 2016, 7:26 PM • 2 Y
Y by Adventure10, Mango247
Here http://www.dms.rs/DMS/data/takmicenja/matematika/osnovna/2016/JSMO16.pdf but on Serbian language.
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mihajlon
374 posts
mihajlon
#4 h  May 21, 2016, 7:58 PM • 2 Y
Y by Adventure10, AlexCenteno2007
Murad.Aghazade wrote:
Hi dude where can I find other Serbian Junior TST problems?

You can find here other Serbia JTST's, but there are not many of those...

PS
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MilosMilicev
241 posts
MilosMilicev
#5 h  Oct 15, 2016, 11:10 AM • 2 Y
Y by Adventure10, Mango247
Let x=our number. For m,n>1, we have that 81Ix, 16Ix, 5Ix, it means that x has at least (4+1)(4+1)(1+1)=50 positive divisors, and for m=1, n=2,x=720(30 positive divisors).It is not hard to prove that it is the minimum number...
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IvanMitrovic
22 posts
IvanMitrovic
#6 h  Mar 8, 2020, 1:51 PM
Y by
MilosMilicev wrote:
Let x=our number. For m,n>1, we have that 81Ix, 16Ix, 5Ix, it means that x has at least (4+1)(4+1)(1+1)=50 positive divisors, and for m=1, n=2,x=720(30 positive divisors).It is not hard to prove that it is the minimum number...

You did not check all of the cases
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Namisgood
30 posts
Namisgood
#7 h  Apr 12, 2025, 1:25 PM
Y by
MilosMilicev wrote:
Let x=our number. For m,n>1, we have that 81Ix, 16Ix, 5Ix, it means that x has at least (4+1)(4+1)(1+1)=50 positive divisors, and for m=1, n=2,x=720(30 positive divisors).It is not hard to prove that it is the minimum number...

sorry I am late



how did you conclude 81|x ?
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