Plan ahead for the next school year. Schedule your class today!

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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
and train with the best! Please note that early bird pricing ends August 19th!
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:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

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0 replies
jwelsh
Jul 1, 2025
0 replies
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
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
Is it true?
lgx57   1
N 42 minutes ago by alexheinis
$0<a_1,a_2\cdots ,a_n$, determine whether it is true.
$$\sum_{i=1}^n \frac{1}{a_i}\ge \sum_{i=1}^n \frac{i}{\sum_{j=1}^i a_j}$$
If not, please give a counterexample.
1 reply
lgx57
Today at 3:02 PM
alexheinis
42 minutes ago
Cone Sul 2020 TST 3 Brazil P2
TiagoCamara   0
2 hours ago
(Cone Sul 2020 TST 3 Brazil P2)Determine all positive integers $n$ for which $4k^2+n$ is a prime number for every $0\leq k< n$ integer.
0 replies
TiagoCamara
2 hours ago
0 replies
Limit of a sequence involving the largest odd divisor
JackMinhHieu   1
N 4 hours ago by mathreyes
Hi everyone,

I came across the following sequence and I’m curious about its behavior:

Let d(k) be the largest odd positive divisor of k. Define a sequence (x_n) by

x_n = (1/n) * sum_{k=1}^{n} (d(k)/k)

Question:
Does the sequence (x_n) converge? If so, what is its limit?

Any insights, proofs, or helpful observations would be appreciated. Thank you!
1 reply
JackMinhHieu
6 hours ago
mathreyes
4 hours ago
IMO 2009, Problem 5
orl   95
N 4 hours ago by mathprodigy2011
Source: IMO 2009, Problem 5
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b + f(a) - 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)

Proposed by Bruno Le Floch, France
95 replies
orl
Jul 16, 2009
mathprodigy2011
4 hours ago
nice inequality by panaitopol
manlio   86
N 4 hours ago by mathprodigy2011
Source: JBMO 2002, Problem 4
Prove that for all positive real numbers $a,b,c$ the following inequality takes place
\[ \frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} . \]
Laurentiu Panaitopol, Romania
86 replies
manlio
Sep 21, 2003
mathprodigy2011
4 hours ago
Chinese Remainder Theorem
MathNerdRabbit103   0
5 hours ago
Hi guys,
Lately i've been trying to understand the proof for the Chinese Remainder Theorem, however i have unfortunately had no luck. Can anybody post about how they understand the proof and please go step by step?
Appreciate it.
0 replies
MathNerdRabbit103
5 hours ago
0 replies
The Second Most Difficult FE in the World
EthanWYX2009   0
Today at 10:43 AM
Source: 2024 March 谜之竞赛-7
Let $\lambda=36+6\sqrt{114}$. Find all functions $f:\mathbb R_+\to\mathbb R_+$, such that for any positive real numbers $x$, $y$,
\[f(x)f(x+\lambda y)f\left(x+\frac 32f(2y)\right)=x(x+\lambda y)f(x+2y)+f(x)f(y)f(x+\lambda y).\]Created by Fanyu Meng
0 replies
EthanWYX2009
Today at 10:43 AM
0 replies
The Most Difficult Functional Equation in the World
EthanWYX2009   0
Today at 10:04 AM
Source: 2023 September 谜之竞赛-3
Determine all functions $f:\mathbb N_+\to\mathbb N_+$, such that for any positive integers $x$, $y$,
\[f(x)^2+y^2\mid\sum_{i=0}^{2023}(xf(x))^{2023-i}\left(f^{(i)}(y)\right)^{2i}.\]Created by Yuxing Ye
0 replies
EthanWYX2009
Today at 10:04 AM
0 replies
Function related to (x+y)^2
61plus   21
N Today at 1:43 AM by player-019
Source: European Girls’ Mathematical Olympiad-2014 - Day 2 - P6
Determine all functions $f:\mathbb R\rightarrow\mathbb R$ satisfying the condition
\[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\]
for all real numbers $x$ and $y$.
21 replies
61plus
Apr 13, 2014
player-019
Today at 1:43 AM
Bounded function satisfying averaging condition
62861   41
N Yesterday at 8:24 PM by ray66
Source: USA Winter Team Selection Test #1 for IMO 2018, Problem 2
Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$,
\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]
Proposed by Yang Liu and Michael Kural
41 replies
62861
Dec 11, 2017
ray66
Yesterday at 8:24 PM
A problem of functional equation
deltapc   3
N Yesterday at 6:28 PM by megarnie
Find all function \( f: \mathbb{R} \to \mathbb{R} \) such as \( f(xf(y^2)+f(x))=f(f(x))f(y)^2+x, \forall x,y \in \mathbb{R} \)
3 replies
deltapc
Yesterday at 2:46 AM
megarnie
Yesterday at 6:28 PM
Max value of function with f(f(n)) < n+50
Rijul saini   3
N Yesterday at 3:14 PM by Saucepan_man02
Source: India IMOTC Day 3 Problem 2
Let $S$ be the set of all non-decreasing functions $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying $f(f(n))<n+50$ for all positive integers $n$. Find the maximum value of
$$f(1)+f(2)+f(3)+\cdots+f(2024)+f(2025)$$over all $f \in S$.

Proposed by Shantanu Nene
3 replies
Rijul saini
Jun 4, 2025
Saucepan_man02
Yesterday at 3:14 PM
easy function equation(question 4)
nima1376   11
N Yesterday at 10:55 AM by MathsII-enjoy
Source: iran tst 2014 third exam
Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that
$x,y\in \mathbb{R}^{+},$ \[ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) \]
11 replies
nima1376
May 21, 2014
MathsII-enjoy
Yesterday at 10:55 AM
Continuity of function and line segment of integer length
egxa   5
N Yesterday at 10:13 AM by GeoGuessr
Source: All Russian 2025 11.8
Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of \( f \). It is known that the graph of \( f \) contains exactly \( N \) chords, one of which has length 2025. Find the minimum possible value of \( N \).
5 replies
egxa
Apr 18, 2025
GeoGuessr
Yesterday at 10:13 AM
Maximum value of function (with two variables)
Saucepan_man02   1
N May 22, 2025 by Saucepan_man02
If $f(\theta) = \min(|2x-7|+|x-4|+|x-2 -\sin \theta|)$, where $x, \theta \in \mathbb R$, then maximum value of $f(\theta)$.
1 reply
Saucepan_man02
May 22, 2025
Saucepan_man02
May 22, 2025
Maximum value of function (with two variables)
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If $f(\theta) = \min(|2x-7|+|x-4|+|x-2 -\sin \theta|)$, where $x, \theta \in \mathbb R$, then maximum value of $f(\theta)$.
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