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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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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   3
N 14 minutes ago by sqing
Let $ a,b>0 $ and $\frac{a}{a^2+3}+ \frac{b}{b^2+ 3} \geq \frac{1}{2} . $ Prove that
$$a^2+ab+b^2\geq 3$$$$a^2-ab+b^2 \geq 1 $$Let $ a,b>0 $ and $\frac{a}{a^3+3}+ \frac{b}{b^3+ 3}\geq \frac{1}{2} . $ Prove that
$$a^3+ab+b^3 \geq 3$$$$ a^3-ab+b^3\geq 1 $$
3 replies
+1 w
sqing
Wednesday at 12:59 PM
sqing
14 minutes ago
J
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Sum of 1/(a^5(b+2c))^2 at least 1/3 [USA TST 2010 2]
MellowMelon   42
N an hour 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
an hour 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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How inflated are current aime/amc problems
derekli   1
N 2 hours ago by Gavin_Deng
So I've been working on a math grinding tool in Stellar Learning (https://stellarlearning.app/competitive) and I was wondering how to make an algorithm that can calculate the difficulty of a problem. Specifically I want to know how difficult past AIMEs and AMC 10s and other contests are, compared to our current contests. I'm planning to make a problem ELO system similar to mathdash or something like that. Any help would be appreciated! Again if you would like to support me you may consider joining our developer team! :D
1 reply
derekli
2 hours ago
Gavin_Deng
2 hours ago
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
J
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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
J
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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 3 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
3 hours ago
J
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exist solutions?
teomihai   6
N 4 hours ago by iwastedmyusername
Find how many perfect squares of five different digits there are, with elements from the set ${0,1,4,6,9}$.
6 replies
teomihai
Yesterday at 5:04 PM
iwastedmyusername
4 hours ago
J
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A pentagon inscribed in a circle of radius √2
tom-nowy   6
N 4 hours ago by anticodon
Can a pentagon with all rational side lengths be inscribed in a circle of radius $\sqrt{2}$ ?
6 replies
tom-nowy
May 6, 2025
anticodon
4 hours ago
J
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J
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Sequence
lgx57   8
N Apr 30, 2025 by Vivaandax
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
8 replies
lgx57
Apr 27, 2025
Vivaandax
Apr 30, 2025
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Sequence
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Reveal topic tags
algebra
Sequences
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G H BBookmark kLocked kLocked NReply
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lgx57
40 posts
lgx57
#1 h  Apr 27, 2025, 12:56 PM
Y by
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
Z K Y
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lgx57
40 posts
lgx57
#2 h  Apr 27, 2025, 12:57 PM
Y by
I can only find that $a_n \sim \sqrt{2n}$.
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aops-g5-gethsemanea2
3456 posts
aops-g5-gethsemanea2
#3 h  Apr 27, 2025, 2:05 PM
Y by
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?
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lgx57
40 posts
lgx57
#4 h  Apr 27, 2025, 2:19 PM
Y by
aops-g5-gethsemanea2 wrote:
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?

Just find a function $f$ ,s.t. $a_n=f(n)$
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steve4916
16 posts
steve4916
#5 h  Apr 27, 2025, 2:24 PM
Y by
now prove me if im wrong but there is no simple closed form for this
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lgx57
40 posts
lgx57
#6 h  Apr 27, 2025, 2:48 PM
Y by
steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?
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johnnie.walker
2 posts
johnnie.walker
#7 h  Apr 29, 2025, 1:42 PM
Y by
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jasperE3
11308 posts
jasperE3
#8 h  Apr 29, 2025, 3:38 PM
Y by
lgx57 wrote:
steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?

why would there be a closed form
Z K Y
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Vivaandax
84 posts
Vivaandax
#9 h  Apr 30, 2025, 5:08 AM
Y by
You can bound the value of a_n quite well (consider IMO Shortlist 1975 Problem 14), but there is not an explicit formula to calculate a_n.
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