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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.

Our full course list for upcoming classes is below:
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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
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IMO 2025 live scoreboards and manifold markets
wnwj   16
N 2 minutes ago by carefully
Live scoreboards:

Part 1
Part 2
Part 3
Part 4

Make predictions at Manifold Markets
16 replies
+16 w
wnwj
Today at 5:18 AM
carefully
2 minutes ago
J
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Inspired by Mexican 2023
sqing   0
9 minutes ago
Source: Own
Let $ a,b>0 ,ab=4 $ and $ b^4+4a^2=16b. $ Prove that
$$a^4+4b^2\geq 16a$$
0 replies
sqing
9 minutes ago
0 replies
J
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JBMO TST Bosnia and Herzegovina 2025 P3
Rotten_   19
N 24 minutes ago by sqing
Let a, b, c be real numbers such that
\[ a + b + c = 0 \quad \text{and} \quad abc = -16. \]Find the minimum value of the expression
\[ W = \frac{a^2 + b^2} {c} + \frac{b^2 + c^2} {a} + \frac{c^2 + a^2} {b}. \]
19 replies
Rotten_
Jul 7, 2025
sqing
24 minutes ago
J
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Interesting Inequality
EthanWYX2009   0
27 minutes ago
Source: 2024 March 谜之竞赛-1
Determine the minimal positive real number \( c \) such that there exists a $2024$-element subset \( A \) of \([6000]\) satisfying the following condition: if we define $T = \{(i, j, k) | i, j, k \in [2000], i + j + k \in A\}$, then for any positive real numbers \( a_1, a_2, \cdots, a_{2000} \),
\[\sum_{(i, j, k) \in T} a_i a_j a_k \leq c \left( \sum_{i=1}^{2000} a_i \right)^3.\]Proposed by Tianqin Li, High School Affiliated to Renmin University of China
0 replies
EthanWYX2009
27 minutes ago
0 replies
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2024 ISL A3
sqing-inequality-BUST   10
N 29 minutes ago by dgrozev
Source: 2024 ISL A3
Decide whether for every sequence $(a_n)$ of positive real numbers,

$\frac{3^{a_1}+3^{a_2}+\cdots+3^{a_n}}{(2^{a_1}+2^{a_2}+\cdots+2^{a_n})^2} < \frac{1}{2024}$

is true for at least one positive integer $n$.
10 replies
2 viewing
sqing-inequality-BUST
Yesterday at 3:02 AM
dgrozev
29 minutes ago
J
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abc=1
hctb00   13
N 30 minutes ago by sqing
Source: unknown
$a,b,c>0,abc=1$,prove:\[\frac{1}{a^2+b+1}+\frac{1}{b^2+c+1}+\frac{1}{c^2+a+1}\le1\]
13 replies
hctb00
Aug 16, 2014
sqing
30 minutes ago
J
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Periodic sequence
EeEeRUT   6
N 34 minutes ago by dgrozev
Source: Isl 2024 A5
Find all periodic sequence $a_1,a_2,\dots$ of real numbers such that the following conditions hold for all $n\geqslant 1$:$$a_{n+2}+a_{n}^2=a_n+a_{n+1}^2\quad\text{and}\quad |a_{n+1}-a_n|\leqslant 1.$$
Proposed by Dorlir Ahmeti, Kosovo
6 replies
EeEeRUT
Yesterday at 3:01 AM
dgrozev
34 minutes ago
J
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Concurrency problem
GGPiku   14
N an hour ago by Rayvhs
Source: Romania 2017 IMO TST 1, problem 1
Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.
14 replies
GGPiku
Mar 18, 2018
Rayvhs
an hour ago
J
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Easy inequality for beginners
kkkdddddwsss   1
N an hour ago by kkkdddddwsss
Prove for any a,b,c,d:
a^2+b^2+c^2+d^2 >= a(b+c+d)
1 reply
kkkdddddwsss
an hour ago
kkkdddddwsss
an hour ago
J
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more incircles and tangents
rmtf1111   87
N an hour ago by AshAuktober
Source: EGMO 2019 P4
Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$
87 replies
1 viewing
rmtf1111
Apr 10, 2019
AshAuktober
an hour ago
J
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FE Tree!
Primeniyazidayi   35
N an hour ago by JARP091
Hello guys!
I want to start a FE tree.One will post a FE equation,and others will try to solve.There will be two problems to solve.If you have solved one of them,post a new one.If two,then post two FEs.
I start:
P1

P2
35 replies
Primeniyazidayi
Jun 12, 2025
JARP091
an hour ago
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IMO 2025 P2
sarjinius   67
N an hour ago by L13832
Source: 2025 IMO P2
Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, so that $C, M, N, D$ lie on $MN$ in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ meets $\Omega$ again at $E\neq A$ and meets $\Gamma$ again at $F\neq A$. Let $H$ be the orthocentre of triangle $PMN$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.

Proposed by Tran Quang Hung, Vietnam
67 replies
+1 w
sarjinius
Jul 15, 2025
L13832
an hour ago
J
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Monstrous FE!
JARP091   6
N an hour ago by math-olympiad-clown
Source: Own
This problem is for anyone who considers themselves a master in FE:
Find all function $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y\in\mathbb{R}$ $x,y$ both not $< 0$,$$f(xf(y)+f(x)f(y))=xf(y)+f(xy).$$
6 replies
JARP091
Jun 30, 2025
math-olympiad-clown
an hour ago
J
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D1052 : How it's possible ?
Dattier   1
N an hour ago by Dattier
Source: les dattes à Dattier
Is it true for all $n$ natural integer : $(E\times 29^n \mod F) \mod 3\neq 0$ ?

E=163999081217965835070356295641931525591357567735624606830696386586994976172813816839262877
50922205042989559280142547393636527346272001339597483126086699049357460700008119117240578043
46281799731794620614941989125738298381362079843446150841376016501310942563338531951229469261
017554376486801


F=666165351866558215458553224258271230186695252725433417706426521946436303489813878149680284
24137540935869820330945301911165461108917069581547697978809314332789769417680417107249591988
71252061235265894516611712110379162326930843580773177773789047845826833190774483296276708089
431095213731040922452939281280
1 reply
Dattier
Tuesday at 4:02 PM
Dattier
an hour ago
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annoying algebra with sequence :/
tabel   2
N Jun 4, 2025 by tabel
Source: random 9th grade text book (section meant for contests)
Let \( a_1 = 1 \) and \( a_{n+1} = 1 + \frac{n}{a_n} \) for \( n \geq 1 \). Prove that the sequence \( (a_n)_{n \geq 1} \) is increasing.
2 replies
tabel
Jun 3, 2025
tabel
Jun 4, 2025
J
annoying algebra with sequence :/
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Reveal topic tags
Sequence
algebra
monotony
L
G H BBookmark kLocked kLocked NReply
Source: random 9th grade text book (section meant for contests)
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tabel
9 posts
tabel
#1 h  Jun 3, 2025, 4:55 PM
Y by
Let \( a_1 = 1 \) and \( a_{n+1} = 1 + \frac{n}{a_n} \) for \( n \geq 1 \). Prove that the sequence \( (a_n)_{n \geq 1} \) is increasing.
Z K Y
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L_.
4 posts
L_.
#2 h  Jun 3, 2025, 10:04 PM
Y by
Notice that, for $n \geq 1$, $$a_{n+2}-a_{n+1}=\frac{n+1}{a_{n+1}}-\frac{n}{a_n}=\frac{n+1}{n}a_n-\frac{n}{a_n}=\frac{na_n^2+a_n^2-n^2}{na_n}$$and that $na_n^2-n^2=na_n >0$, this concludes the proof.
Z K Y
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tabel
9 posts
tabel
#4 h  Jun 4, 2025, 4:50 AM
Y by
L_. wrote:
Notice that, for $n \geq 1$, $$a_{n+2}-a_{n+1}=\frac{n+1}{a_{n+1}}-\frac{n}{a_n}=\frac{n+1}{n}a_n-\frac{n}{a_n}=\frac{na_n^2+a_n^2-n^2}{na_n}$$and that $na_n^2-n^2=na_n >0$, this concludes the proof.

It's wrong , you misunderstood the recurrence relation .
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