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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
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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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AOPS MO Introduce
MathMaxGreat   31
N 4 minutes ago by Royal_mhyasd
$AOPS MO$

Problems: post it as a private message to me or @jerryZYang, please post it in $LATEX$ and have answers

6 Problems for two rounds, easier than $IMO$

If you want to do the problems or be interested, reply ’+1’
Want to post a problem reply’+2’ and message me
Want to be in the problem selection committee, reply’+3’
31 replies
MathMaxGreat
Today at 1:04 AM
Royal_mhyasd
4 minutes ago
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Analytic NT on Polynomials with algebra assistance
Assassino9931   11
N 8 minutes ago by Oksutok
Source: RMM 2024 Problem 6
A polynomial $P$ with integer coefficients is square-free if it is not expressible in the form $P = Q^2R$, where $Q$ and $R$ are polynomials with integer coefficients and $Q$ is not constant. For a positive integer $n$, let $P_n$ be the set of polynomials of the form
$$1 + a_1x + a_2x^2 + \cdots + a_nx^n$$with $a_1,a_2,\ldots, a_n \in \{0,1\}$. Prove that there exists an integer $N$ such that for all integers $n \geq N$, more than $99\%$ of the polynomials in $P_n$ are square-free.

Navid Safaei, Iran
11 replies
Assassino9931
Feb 29, 2024
Oksutok
8 minutes ago
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D1033 : A problem of probability for dominoes 3*1
Dattier   4
N 9 minutes ago by Dattier
Source: les dattes à Dattier
Let $G$ a grid of 9*9, we choose a little square in $G$ of this grid three times, we can choose three times the same.

What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
4 replies
Dattier
May 15, 2025
Dattier
9 minutes ago
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Subsets disjoint with colours
Twoisaprime   11
N 11 minutes ago by Oksutok
Source: RMM 2025 P6
Let $k$ and $m$ be integers greater than $1$. Consider $k$ pairwise disjoint sets $S_1,S_2, \cdots S_k$; each of these sets has exactly $m+1$ elements, one of which is red and the other $m$ are all blue. Let $\mathcal{F}$ be the family of all subsets $F$ of $S_1 \bigcup S_2\bigcup \cdots S_k$ such that, for every $i$ , the intersection $F \bigcap S_i$ is monochromatic; the empty set is also monochromatic. Determine the largest cardinality of a subfamily $\mathcal{G} \subseteq \mathcal{F}$, no two sets of which are disjoint.
Proposed by Russia, Andrew Kupavskii and Maksim Turevskii
11 replies
Twoisaprime
Feb 13, 2025
Oksutok
11 minutes ago
J
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Great Inequality
EthanWYX2009   2
N 13 minutes ago by Photaesthesia
Source: 2025 June 谜之竞赛-3
Given positive integer \( n \geq 2 \) and positive real \( t \). Let positive real numbers \( a_1, a_2, \ldots, a_n \) satisfy \( \sum_{i=1}^n a_i = t \). Denote \( S = \{1, 2, \cdots, n\} \). A non-empty subset \( I \) of \( S \) is called good if
\[\sum_{i\in I} a_i^3 \geq \left( \sum_{i\in I} a_i \right)^2.\]Determine the maximum possible number of good subsets of \( S \).

Created by Yuxing Ye
2 replies
EthanWYX2009
4 hours ago
Photaesthesia
13 minutes ago
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IMO Genre Predictions
ohiorizzler1434   114
N 33 minutes ago by Shan3t
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
114 replies
ohiorizzler1434
May 3, 2025
Shan3t
33 minutes ago
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Nice 60 degree triangle property
Kratsneb   1
N an hour ago by RANDOM__USER
Let $ABC$ be a triangle with $\angle A = 60 ^{\circ}$. $AD$, $BE$ and $CF$ are the altitudes, $M$ is the midpoint of $BC$. $BE \cap (ABC) = P$, $CF \cap (ABC) = Q$. Let $I$ and $J$ be incenters of $\triangle CEP$ and $\triangle BFQ$. Prove that $I$, $J$, $D$, $M$ lie on a circle.
IMAGE
It works for $\angle A = 120 ^{\circ}$ too.
1 reply
Kratsneb
Thursday at 2:13 PM
RANDOM__USER
an hour ago
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Bashy chessboard problem
BR1F1SZ   1
N an hour ago by Gggvds1
Source: 2024 May Olympiad L2 P1
A $4\times 8$ grid is divided into $32$ unit squares. There are square tiles of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$ and $4 \times 4$. The goal is to completely cover the grid using exactly $n$ of these tiles.
[list=a]
[*]Is it possible to do this if $n = 19$?
[*]Is it possible to do this if $n = 14$?
[*]Is it possible to do this if $n = 7$?
[/list]
Note: The tiles cannot overlap or extend beyond the grid.
1 reply
BR1F1SZ
Jan 25, 2025
Gggvds1
an hour ago
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AOPS MO problem selection committee selection
MathMaxGreat   7
N an hour ago by Jackson0423
Problems to select the one that will be in the AOPS MO problem selection committee

(All old problems but I will not tell you were)
7 replies
1 viewing
MathMaxGreat
2 hours ago
Jackson0423
an hour ago
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Extremum problem
Virgil Nicula   1
N an hour ago by Mathzeus1024
Source: Crux Mathematicorum
Ascertain $\min\left\{x+y+\sqrt{x^2+y^2}\right\}$ , where $a>0$ , $b>0$ are given and

$\left\|\begin{array}{c} x>a\ \wedge\ y>b\\\\ \frac ax\ +\ \frac by\ =\ 1\end{array}\right\|$ . Present (if is possibly) a geometrical interpretation.
1 reply
Virgil Nicula
Aug 4, 2013
Mathzeus1024
an hour ago
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f(xf(y)-yf(x))=f(xy)-xf(y)
ChildFlower   1
N an hour ago by RagvaloD
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ we have:
$$f(xf(y)-yf(x))=f(xy)-xf(y)$$
1 reply
ChildFlower
3 hours ago
RagvaloD
an hour ago
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Parallel lines (extension of previous problem)
RANDOM__USER   3
N an hour ago by Lil_flip38
Source: Own
Let \(D\) be an arbitrary point on the side \(BC\) in a triangle \(\triangle{ABC}\). Let \(E\) and \(F\) be the intersection of the lines parallel to \(AC\) and \(AB\) through \(D\) with \(AB\) and \(AC\). Let \(G\) be the intersection of \((AFE)\) with \((ABC)\). Let \(M\) be the midpoint of \(BC\) and \(X\) the intersection of \(AM\) with \((ABC)\). Let \(H\) be the intersection of \((XMG)\) with \(BC\). Prove that \(EF\) is parallel to \(AH\).

IMAGE

Note: This is another property of a configuration I posted before where one needed to prove that \(X, D\) and \(G\) are collinear. There are surprisingly many properties in the configuration posted earlier :P
3 replies
RANDOM__USER
6 hours ago
Lil_flip38
an hour ago
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inequalities
pennypc123456789   6
N 2 hours ago by ohiorizzler1434
If $a,b,c$ are positive real numbers, then
$$ \frac{a + b}{a + 7b + c} + \dfrac{b + c}{b + 7c + a}+\dfrac{c + a}{c + 7a + b} \geq \dfrac{2}{3}$$
we can generalize this problem
6 replies
pennypc123456789
Apr 17, 2025
ohiorizzler1434
2 hours ago
J
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Constructing a Sequence with Density 1/lnn
EthanWYX2009   0
2 hours ago
Source: 2025 May 谜之竞赛-6
Prove that there exists a strictly increasing infinite sequence of positive integers \(\{a_n\}\) satisfying the following two conditions: [list]
[*]For any positive integers \(x \leq y < z\), the following inequality holds:
\[ \sum_{n=x}^{y} a_n \neq a_z; \][*]There exist a positive constant \(c\) and a positive integer \(M\) such that for any integer \(m > M\), the set \(\{1, 2, \cdots, m\}\) contains at least \(\frac{cm}{\ln m}\) elements appearing in the sequence \(\{a_n\}\).
[/list]
Created by Zheng Yang
0 replies
EthanWYX2009
2 hours ago
0 replies
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one variable function
youochange   1
N May 17, 2025 by Fishheadtailbody
$f:\mathbb R-\{0,1\} \to \mathbb R$


$f(x)+f(\frac{1}{1-x})=2x$
1 reply
youochange
May 17, 2025
Fishheadtailbody
May 17, 2025
J
one variable function
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Reveal topic tags
function
functional equation
algebra
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youochange
194 posts
youochange
#1 h  May 17, 2025, 2:49 PM • 1 Y
Y by PikaPika999
$f:\mathbb R-\{0,1\} \to \mathbb R$


$f(x)+f(\frac{1}{1-x})=2x$
Z K Y
The post below has been deleted. Click to close.
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Fishheadtailbody
9 posts
Fishheadtailbody
#2 h  May 17, 2025, 3:13 PM • 1 Y
Y by youochange
Notice that if $g(x) = \frac{1}{1 - x}$, then $g^3(x) = g(x)$.
Hence we can consider the following, which is simply a system of equations with three variables,
\begin{align*} f(x) + f(g(x)) = 2x, \\ f(g(x)) + f(g(g(x)) = 2g(x), \\ f(g^2(x)) + f(x) = 2g^2(x). \end{align*}Thus,
$f(x) = x - g(x) + g^2(x) = x - \frac{1}{1 - x} + \frac{x - 1}{x}$.
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