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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

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0 replies
jwelsh
Jul 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
J
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Goals for 2025-2026
Airbus320-214   395
N 8 minutes ago by sadas123
Please write down your goal/goals for competitions here for 2025-2026.
395 replies
Airbus320-214
May 11, 2025
sadas123
8 minutes ago
J
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Find limit of sequence
williamphan913   1
N 18 minutes ago by Ikromjon
Sequence $(a_n)$ is defied by:
$a_1=1; a_{n+1}=\frac{1}{(2n+1)a_n}, \forall n=1, 2, 3,... $
a/ Prove that $(a_n)$ has a finite limit, find that limit.
b/ Prove that: $a_1+a_2+...+a_n+1>\sqrt{2n+1}, \forall n=1, 2, 3,...$
1 reply
williamphan913
Nov 1, 2023
Ikromjon
18 minutes ago
J
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Strip region cover
c7h5n3o6_tnt   0
21 minutes ago
Let \( S \) be an infinite set of points on a plane. It is known that any 3 points in \( S \) can be covered by a strip region with a width of $1$. Prove that \( S \) can be covered by a strip region with a width of $2$.
0 replies
c7h5n3o6_tnt
21 minutes ago
0 replies
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(x + y)(y + z)(z + x)/{xyz} if (x+ y}/z=(y + z)/x=(z + x)/y
parmenides51   9
N 28 minutes ago by neeyakkid23
Source: 2023 Austrian Mathematical Olympiad, Junior Regional Competition , Problem 1
Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$
(Walther Janous)
9 replies
parmenides51
Mar 26, 2024
neeyakkid23
28 minutes ago
J
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Interesting Succession
AlexCenteno2007   4
N 32 minutes ago by AlexCenteno2007
The sequence $\{a_n\}$ of integers is defined by
\[ -\frac{1}{2} < a_{n+1} - \frac{a_n^2}{a_{n-1}} \leq \frac{1}{2} \]with $a_1 = 2$, $a_2 = 7$, prove that $a_n$ is odd for all values of $n \geq 2$.
4 replies
AlexCenteno2007
Yesterday at 6:21 PM
AlexCenteno2007
32 minutes ago
J
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Looks like TST 2000
v_Enhance   41
N 36 minutes ago by L13832
Source: USA January TST for IMO 2012, Problem 2
In cyclic quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $P$. Let $E$ and $F$ be the respective feet of the perpendiculars from $P$ to lines $AB$ and $CD$. Segments $BF$ and $CE$ meet at $Q$. Prove that lines $PQ$ and $EF$ are perpendicular to each other.
41 replies
v_Enhance
Aug 23, 2013
L13832
36 minutes ago
J
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Weird but silly functional inequality
MarkBcc168   10
N 44 minutes ago by math-olympiad-clown
Source: 2020 Thailand Mathematical Olympiad P7
Determine all functions $f:\mathbb{R}\to\mathbb{Z}$ satisfying the inequality $(f(x))^2+(f(y))^2 \leq 2f(xy)$ for all reals $x,y$.
10 replies
MarkBcc168
Dec 28, 2020
math-olympiad-clown
44 minutes ago
J
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Interesting
AlexCenteno2007   1
N an hour ago by AlexCenteno2007
For positive real numbers \( x_1, x_2, \dots, x_{n+1} \), prove that
\[ \frac{1}{x_1} + \frac{x_1}{x_2} + \frac{x_1 x_2}{x_3} + \cdots + \frac{x_1 x_2 \cdots x_n}{x_{n+1}} \geq 4\left(1 - x_1 x_2 \cdots x_{n+1}\right). \]
1 reply
AlexCenteno2007
Today at 2:33 AM
AlexCenteno2007
an hour ago
J
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Reversing a long snake
62861   12
N an hour ago by sato2718
Source: USA Winter TST for IMO 2019, Problem 3, by Nikolai Beluhov
A snake of length $k$ is an animal which occupies an ordered $k$-tuple $(s_1, \dots, s_k)$ of cells in a $n \times n$ grid of square unit cells. These cells must be pairwise distinct, and $s_i$ and $s_{i+1}$ must share a side for $i = 1, \dots, k-1$. If the snake is currently occupying $(s_1, \dots, s_k)$ and $s$ is an unoccupied cell sharing a side with $s_1$, the snake can move to occupy $(s, s_1, \dots, s_{k-1})$ instead. The snake has turned around if it occupied $(s_1, s_2, \dots, s_k)$ at the beginning, but after a finite number of moves occupies $(s_k, s_{k-1}, \dots, s_1)$ instead.

Determine whether there exists an integer $n > 1$ such that: one can place some snake of length $0.9n^2$ in an $n \times n$ grid which can turn around.

Nikolai Beluhov
12 replies
62861
Dec 10, 2018
sato2718
an hour ago
J
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Elegant Geometry
EthanWYX2009   0
an hour ago
Source: 2024 May 谜之竞赛-4
In the triangle \(ABC\), let \(I\) be the incenter. Denote \(X\) as the intersection of \(BC\) and the perpendicular bisector of \(AI\), \(Y\) as the intersection of \(CA\) and the perpendicular bisector of \(BI\), and \(Z\) as the intersection of \(AB\) and the perpendicular bisector of \(CI\).

Prove that the circumcircles of \(\triangle AIX\), \(\triangle BIY\), and \(\triangle CIZ\) are coaxial.

Proposed by Yuan Zhang, Shijiazhuang No. 2 High School
0 replies
EthanWYX2009
an hour ago
0 replies
J
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harder than ever
perfect_square   2
N an hour ago by perfect_square
Let $a,b,c$ which satisfy:
$ \begin{cases} a+b+c=4 \\ ab+bc+ca=5 \end{cases} $
and $w=abc$
a. Prove that: $ \frac{50}{27} \le w \le 2$
b. Given $ n \in N, n \ge 3$. Prove that: $ a^n+b^n+c^n=f(w)$, which is increasing function.
2 replies
perfect_square
Jul 16, 2025
perfect_square
an hour ago
J
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OpenAI won gold on 2025 IMO
centslordm   129
N 4 hours ago by Demetri
it got 35/42
3 years ago it got 2; aura imo

[its solutions]
129 replies
1 viewing
centslordm
Jul 20, 2025
Demetri
4 hours ago
J
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FREE AMC 10 Bootcamp - Shine4Love Teens Club
mop   27
N 6 hours ago by senboy
Hey everyone!

Want to get a head-start with AMC prep this year and bag that AIME qual/(D)HR this November? The Shine4Love Teens Club is excited to host a FREE 3 week long AMC 10 Bootcamp starting next Monday!

The course will cover fundamental concepts and problem-solving strategies for the AMC 10 in algebra, geometry, combinatorics, and number theory, providing the experience needed to give you the best shot at accomplishing your AMC goals!

Registration is completely free, although donations are open (all proceeds will be donated to charity!).

Date/Time:
4:00-6:00 p.m. PST on Mondays, Wednesdays, & Fridays

Format:
Classes will be held on Zoom, and class notes will be posted after each class on Google Classroom. Additional problem sets will be posted on Google Classroom for students to work on. The instructor's email will be open 24/7 for any questions!

Register/more details at tinyurl.com/amc10prepcourse

Hope to see you all soon!
27 replies
mop
Jul 15, 2025
senboy
6 hours ago
J
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Does anyone have access to the current AMM edition? I’d like to check out the pr
Khalifakhalifa   1
N Today at 4:36 AM by Khalifakhalifa
Does anyone have access to the current AMM edition? I’d like to check out the problems section.
1 reply
Khalifakhalifa
Yesterday at 11:15 AM
Khalifakhalifa
Today at 4:36 AM
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J
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Zsigmondy's theorem
V0305   24
N Jun 19, 2025 by Mr.Sharkman
Is Zsigmondy's theorem allowed on the IMO, and is it allowed on the AMC series of proof competitions (e.g. USAJMO, USA TSTST)?
24 replies
V0305
May 24, 2025
Mr.Sharkman
Jun 19, 2025
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Zsigmondy's theorem
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Reveal topic tags
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V0305
746 posts
V0305
#1 h  May 24, 2025, 6:22 PM • 1 Y
Y by GA34-261
Is Zsigmondy's theorem allowed on the IMO, and is it allowed on the AMC series of proof competitions (e.g. USAJMO, USA TSTST)?
Z K Y
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vincentwant
1496 posts
vincentwant
#2 h  May 24, 2025, 6:24 PM • 1 Y
Y by GA34-261
yes it is allowed (tstst 2018/8 i think requires it)
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Andyluo
1072 posts
Andyluo
#3 h  May 24, 2025, 6:39 PM • 1 Y
Y by GA34-261
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

I was curious about citing sources in general.

can you just site anything thats from a book or smth?
Z K Y
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CatCatHead
44 posts
CatCatHead
#4 h  May 25, 2025, 9:00 AM • 1 Y
Y by GA34-261
It is allowed and very useful in some special problems
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N3bula
316 posts
N3bula
#5 h  May 25, 2025, 11:01 PM • 1 Y
Y by GA34-261
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

Nothing will ever require zsigmondy in a well made contest, note stuff like china contests/rmm are not well made olympiads. The problem you mentioned does not require zsigmondy. Also in terms of citing results if its a named result its citable, be careful with citing stuff from muricaaaa or such tho and it very much depends on what country ur from for what is citable.
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EGMO
160 posts
EGMO
#6 h  May 25, 2025, 11:36 PM • 1 Y
Y by GA34-261
buh china contests are decent :(
dissing my home country is wild
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N3bula
316 posts
N3bula
#7 h  May 25, 2025, 11:39 PM • 1 Y
Y by GA34-261
EGMO wrote:
buh china contests are decent :(
dissing my home country is wild

China contests are not good or accurate olympiad contests
Z K Y
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ohiorizzler1434
946 posts
ohiorizzler1434
#8 h  May 25, 2025, 11:50 PM • 2 Y
Y by Andyluo, GA34-261
buh we should analyse each contest according to each situation! china contests are unique in their own way and we should not disrespect it for having bad problems!
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pingpongmerrily
4055 posts
pingpongmerrily
#9 h  May 26, 2025, 12:05 AM • 1 Y
Y by GA34-261
I'm confused, what is special about Zsigmondy's theorem which makes it different from other theorems in terms of being cited? Why can't you just cite it like any other theorem such as the pythagorean theorem or fermat's little theorem or whatever
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vincentwant
1496 posts
vincentwant
#10 h  May 26, 2025, 12:29 AM • 2 Y
Y by aidan0626, GA34-261
pingpongmerrily wrote:
I'm confused, what is special about Zsigmondy's theorem which makes it different from other theorems in terms of being cited? Why can't you just cite it like any other theorem such as the pythagorean theorem or fermat's little theorem or whatever

probably because it is very hard to prove
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imagien_bad
77 posts
imagien_bad
#11 h  May 26, 2025, 12:29 AM • 2 Y
Y by bachkieu, GA34-261
xig $ $
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N3bula
316 posts
N3bula
#12 h  May 26, 2025, 1:27 AM • 1 Y
Y by GA34-261
pingpongmerrily wrote:
I'm confused, what is special about Zsigmondy's theorem which makes it different from other theorems in terms of being cited? Why can't you just cite it like any other theorem such as the pythagorean theorem or fermat's little theorem or whatever

Harder to prove but not stupid stupid hard either tho
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CatCatHead
44 posts
CatCatHead
#13 h  May 26, 2025, 1:54 AM • 1 Y
Y by GA34-261
Andyluo wrote:
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

I was curious about citing sources in general.

can you just site anything thats from a book or smth?

Actually in the TST level you can cite anything if you remember the right place like in lemma 4.13 in the EGMO book written by evan chen balabala. But in the USAJMO/USAMO level you may be marked wrong as there are too much students, so I do not recommond to use not famous lemma in USAJMO and USAMO. (Zsigmondy is famous enough i think)
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N3bula
316 posts
N3bula
#14 h  May 26, 2025, 3:51 AM • 1 Y
Y by GA34-261
CatCatHead wrote:
Andyluo wrote:
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

I was curious about citing sources in general.

can you just site anything thats from a book or smth?

Actually in the TST level you can cite anything if you remember the right place like in lemma 4.13 in the EGMO book written by evan chen balabala. But in the USAJMO/USAMO level you may be marked wrong as there are too much students, so I do not recommond to use not famous lemma in USAJMO and USAMO. (Zsigmondy is famous enough i think)

Note that citing aops handouts does not count, if its a published book yes an aops handout so stuff like muricaaaa I dont believe is qoutable.
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Martin2001
175 posts
Martin2001
#15 h  May 26, 2025, 2:38 PM • 1 Y
Y by GA34-261
Zsig is most definitely allowed, i have used it many times. Remember that the people grading the usajmo/amo and so on are themselves very orz so relatively well known things like Zsig can be used.
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llddmmtt1
433 posts
llddmmtt1
#17 h  May 26, 2025, 8:50 PM • 3 Y
Y by aidan0626, GA34-261, MathRook7817
what the zsigma
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dolphinday
1335 posts
dolphinday
#18 h  May 27, 2025, 1:57 AM • 1 Y
Y by GA34-261
you can use any well known result for olympiad contests. Zsig isn't that obscure
it's also used for 2000 imo p5
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whwlqkd
184 posts
whwlqkd
#19 h  May 27, 2025, 7:22 AM • 1 Y
Y by GA34-261
Everything is allowed if you mention the theorem name and statement.
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Stead
12 posts
Stead
#20 h  Jun 3, 2025, 11:08 AM
Y by
I suppose then, if Fermat's Last Theorem would somehow help you make progress in a problem, then it is allowed?
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CatCatHead
44 posts
CatCatHead
#21 h  Jun 3, 2025, 11:50 AM
Y by
Stead wrote:
I suppose then, if Fermat's Last Theorem would somehow help you make progress in a problem, then it is allowed?

yeah， actually you can use it in USA TSTST 2024 Q6
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Stead
12 posts
Stead
#22 h  Jun 3, 2025, 1:52 PM
Y by
Oh, nice! Will take a look. Thanks!
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Sudokan
1 post
Sudokan
#23 h  Jun 3, 2025, 2:18 PM
Y by
What exactly is this theorem? Thank you!
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CatCatHead
44 posts
CatCatHead
#24 h  Jun 4, 2025, 12:35 AM
Y by
Sudokan wrote:
What exactly is this theorem? Thank you!

zsigmondy means that you can always find new primes from a^n-b^n when n change to n+1 every time except fro two specail cases.
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Mr.Sharkman
589 posts
Mr.Sharkman
#25 h  Jun 19, 2025, 5:35 PM
Y by
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

China TST 2005/9
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Mr.Sharkman
589 posts
Mr.Sharkman
#26 h  Jun 19, 2025, 5:36 PM
Y by
CatCatHead wrote:
Stead wrote:
I suppose then, if Fermat's Last Theorem would somehow help you make progress in a problem, then it is allowed?

yeah， actually you can use it in USA TSTST 2024 Q6

OH YEAH LOL

That may be the weirded fe ever
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