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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
J
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Monstrous FE!
JARP091   6
N 13 minutes 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
13 minutes ago
J
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D1052 : How it's possible ?
Dattier   1
N 19 minutes 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
19 minutes ago
J
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Play in intergers
AbdulWaheed   2
N 20 minutes ago by CHESSR1DER
Find all positive integers p such that $\frac{(p-1)!+p+1}{p(p+1)}\in\mathbb{Z}.$
2 replies
AbdulWaheed
2 hours ago
CHESSR1DER
20 minutes ago
J
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Side length geo
EeEeRUT   9
N 28 minutes ago by Diamond-jumper76
Source: Isl 2024 G8
Let $ABC$ be a triangle with $AB<AC<BC$, and let $D$ be a point in the interior of segment $BC$. Let $E$ be a point on the circumcircle of triangle $ABC$ such that $A$ and $E$ lie on opposite sides of the line $BC$ and $\angle{BAD}=\angle{EAC}$. Let $I,I_B,I_C,J_B$ and $J_C$ be the incenters of triangles $ABC,ABD,ADC,ABE$, and $AEC$, respectively. Prove that $I_B,I_C,J_B$, and $J_C$ are concyclic if and only if $AI,I_BJ_C$, and $J_BI_C$ concur.
9 replies
EeEeRUT
Yesterday at 3:01 AM
Diamond-jumper76
28 minutes ago
J
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2024 SL C5
Twoisaprime   5
N 31 minutes ago by Diamond-jumper76
Source: 2024 IMO Shortlist C5
Let $N$ be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers $1, 2, \dots, N$ on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers $(k, n)$, where $k \geq 0$ and $n$ is one of the integers on the board, and then erasing every integer $s$ on the board such that $2^k \mid n - s$. The game continues until the board is empty. The player who erases the last integer on the board loses.

Determine all values of $N$ for which Geoff can ensure that he wins, no matter how Ceri plays.
5 replies
Twoisaprime
Yesterday at 3:03 AM
Diamond-jumper76
31 minutes ago
J
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Midpoint of arc black magic
MarkBcc168   11
N 37 minutes ago by Diamond-jumper76
Source: IMO Shortlist 2024 G7
Let \(ABC\) be a triangle with incenter \(I\) such that \(AB<AC<BC\). The second intersections of \(AI\), \(BI\), and \(CI\) with the circumcircle of triangle \(ABC\) are \(M_{A}\), \(M_{B}\), and \(M_{C}\), respectively. Lines \(AI\) and \(BC\) intersect at \(D\) and lines \(BM_{C}\) and \(CM_{B}\) intersect at \(X\). Suppose the circumcircle of triangles \(XM_{B}M_{C}\) and \(XBC\) intersect again at \(S\neq X\). Lines \(BX\) and \(CX\) intersect the circumcircle of triangle \(SXM_{A}\) again at \(P\neq X\) and \(Q\neq X\), respectively.

Prove that the circumcenter of triangle \(SID\) lies on \(PQ\).

Proposed by Thailand
11 replies
MarkBcc168
Yesterday at 3:01 AM
Diamond-jumper76
37 minutes ago
J
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Hard sequence
straight   1
N 40 minutes ago by straight
Source: Own
Consider a sequence $(a_n)_n, n \rightarrow \infty$ of real numbers.

Consider an infinite $\mathbb{N} \times \mathbb{N}$ grid $a_{i,j}$. In the first row of this grid, we place $a_0$ in every square ($a_{0,n} = a_0)$. In the first column of this grid, we place $a_n$ in the $n$-th square ($a_{n,0} = a_n)$.
Next, fill up the grid according to the following rule: $a_{i,j} = a_{i-1,j} + a_{i,j-1}$.

If $\lim_{i \rightarrow \infty} a_{i,j} = \infty$ for all $j = 0,1,...$, does this mean that $a_n = 0$ for all $n$?

Hint?
1 reply
straight
Yesterday at 10:59 PM
straight
40 minutes ago
J
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I am [not] a parallelogram
peppapig_   14
N 40 minutes ago by Diamond-jumper76
Source: ISL 2024/G4
Let $ABCD$ be a quadrilateral with $AB$ parallel to $CD$ and $AB<CD$. Lines $AD$ and $BC$ intersect at a point $P$. Point $X$ distinct from $C$ lies on the circumcircle of triangle $ABC$ such that $PC=PX$. Point $Y$ distinct from $D$ lies on the circumcircle of triangle $ABD$ such that $PD=PY$. Lines $AX$ and $BY$ intersect at $Q$.

Prove that $PQ$ is parallel to $AB$.

Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine
14 replies
peppapig_
Yesterday at 3:00 AM
Diamond-jumper76
40 minutes ago
J
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IMO 2025 live scoreboards and manifold markets
wnwj   11
N 41 minutes ago by Royal_mhyasd
Live scoreboards:

Part 1
Part 2
Part 3
Part 4

Make predictions at Manifold Markets
11 replies
+1 w
wnwj
Today at 5:18 AM
Royal_mhyasd
41 minutes ago
J
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quadrilateral geo with length conditions
OronSH   12
N 44 minutes ago by MathsII-enjoy
Source: IMO Shortlist 2024 G1
Let $ABCD$ be a cyclic quadrilateral such that $AC<BD<AD$ and $\angle DBA<90^\circ$. Point $E$ lies on the line through $D$ parallel to $AB$ such that $E$ and $C$ lie on opposite sides of line $AD$, and $AC=DE$. Point $F$ lies on the line through $A$ parallel to $CD$ such that $F$ and $C$ lie on opposite sides of line $AD$, and $BD=AF$.

Prove that the perpendicular bisectors of segments $BC$ and $EF$ intersect on the circumcircle of $ABCD$.

Proposed by Mykhailo Shtandenko, Ukraine
12 replies
OronSH
Yesterday at 3:13 AM
MathsII-enjoy
44 minutes ago
J
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about a question
Froster   0
3 hours ago
Request for Help with This Integral on Aops Online

Hey everyone on Aops Online!

I'm really stuck on computing this integral:

\int_{-a}^{a} \frac{\sqrt{a^4 + (a^2 + b^2)x^2}}{a\sqrt{a^2 + x^2}} \, dx

where a and b are constants. I've tried simplifying the integrand, using substitution (like letting x = a\tan\theta for the \sqrt{a^2 + x^2} part), and checking if it's an even function to simplify the interval (since replacing x with -x keeps the integrand the same, so it becomes 2\int_{0}^{a} \cdots dx). But I still can't get to a clear result.

Has anyone dealt with a similar integral before? Any tips on substitution, simplifying the square - root terms, or recognizing a standard integral form here? Thanks a ton for your help!
0 replies
Froster
3 hours ago
0 replies
J
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Ez inequality with AM-GM
Tofa7a._.36   4
N 3 hours ago by DAVROS
Prove that for all positive real numbers $x,y,z$ such that $xy+yz+zx = 3$.
We have:
$$\dfrac{1}{xyz} \ge \dfrac{2}{3}(\sqrt{x} + \sqrt{y} + \sqrt{z}) -1$$
4 replies
Tofa7a._.36
Jul 15, 2025
DAVROS
3 hours ago
J
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AM-GM Problem
arcticfox009   19
N 5 hours ago by sqing
Let $x, y$ be positive real numbers such that $xy \geq 1$. Find the minimum value of the expression

\[ \frac{(x^2 + y)(x + y^2)}{x + y}. \]
answer confirmation
19 replies
arcticfox009
Jul 11, 2025
sqing
5 hours ago
J
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Inequalities
sqing   16
N 5 hours ago by sqing
Let $ a,b,c $ be real numbers . Prove that
$$- \frac{64(9+2\sqrt{21})}{9} \leq \frac {(ab-4)(bc-4)(ca-4) } {(a^2+a +1)(b^2+b +1)(c^2+c +1)}\leq \frac{16}{9}$$$$- \frac{8(436+79\sqrt{31})}{27} \leq \frac {(ab-5)(bc-5)(ca-5) } {(a^2+a +1)(b^2+b +1)(c^2+c +1)}\leq \frac{25}{12}$$
16 replies
sqing
Jul 6, 2025
sqing
5 hours ago
J
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J
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Compilation of functions problems
Saucepan_man02   5
N May 12, 2025 by Konigsberg
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..
5 replies
Saucepan_man02
May 7, 2025
Konigsberg
May 12, 2025
J
Compilation of functions problems
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Reveal topic tags
function
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Saucepan_man02
1401 posts
Saucepan_man02
#1 h  May 7, 2025, 6:15 AM
Y by
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..
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derekli
57 posts
derekli
#2 h  May 7, 2025, 8:14 PM
Y by
Saucepan_man02 wrote:
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..

Once the stellar learning tagging system is complete (stellarlearning.app/competitive), we will have the ability to practice any problems and filter out by topics.
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Saucepan_man02
1401 posts
Saucepan_man02
#3 h  May 10, 2025, 12:45 AM
Y by
@above Kindly update here when its done :coolspeak: .

Could anyone kindly post some handout/compilation of problems for functions meanwhile?
Z K Y
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Saucepan_man02
1401 posts
Saucepan_man02
#4 h  May 12, 2025, 3:29 AM
Y by
\bump 8char
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lightsbug
8 posts
lightsbug
#5 h  May 12, 2025, 6:39 AM
Y by
What kind of function-related problems are you looking for? You might find something here.
Z K Y
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Konigsberg
2240 posts
Konigsberg
#6 h  May 12, 2025, 7:16 PM
Y by
Any resource under “Algebra” here should have a chapter on functions: https://tinyurl.com/ContestGuideIntlGDrive

One resource in particular linked there, the “AoPS MegaStore”, has a lot of handouts sorted by subject.
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