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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 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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number theory
frost23   1
N 4 minutes ago by frost23
Source: own
is it true that al numbers of the form $a1a1a2a2.........anan$ which is a perfect square is of the form $10^2(n-2).88^2$
1 reply
frost23
5 minutes ago
frost23
4 minutes ago
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Integer polynomial commutes with sum of digits
cjquines0   45
N 9 minutes ago by anudeep
Source: 2016 IMO Shortlist N1
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
Proposed by Warut Suksompong, Thailand
45 replies
cjquines0
Jul 19, 2017
anudeep
9 minutes ago
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Good Permutations in Modulo n
swynca   11
N 18 minutes ago by Assassino9931
Source: BMO 2025 P1
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
11 replies
swynca
Apr 27, 2025
Assassino9931
18 minutes ago
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Israel Number Theory
mathisreaI   67
N 29 minutes ago by lolsamo
Source: IMO 2022 Problem 5
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
67 replies
mathisreaI
Jul 13, 2022
lolsamo
29 minutes ago
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IMO 2016 Shortlist, N6
dangerousliri   69
N 33 minutes ago by MR.1
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.

Proposed by Dorlir Ahmeti, Albania
69 replies
dangerousliri
Jul 19, 2017
MR.1
33 minutes ago
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Number of Solutions is 2
Miku3D   30
N an hour ago by lakshya2009
Source: 2021 APMO P1
Prove that for each real number $r>2$, there are exactly two or three positive real numbers $x$ satisfying the equation $x^2=r\lfloor x \rfloor$.
30 replies
Miku3D
Jun 9, 2021
lakshya2009
an hour ago
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Problem 3
blug   3
N an hour ago by sunken rock
Source: Czech-Polish-Slovak Junior Match 2025 Problem 3
In a triangle $ABC$, $\angle ACB=60^{\circ}$. Points $D, E$ lie on segments $BC, AC$ respectively. Points $K, L$ are such that $ADK$ and $BEL$ are equlateral, $A$ and $L$ lie on opposite sides of $BE$, $B$ and $K$ lie on the opposite siedes of $AD$. Prove that
$$AE+BD=KL.$$
3 replies
blug
Yesterday at 4:47 PM
sunken rock
an hour ago
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Computing functions
BBNoDollar   6
N an hour ago by BBNoDollar
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[ f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0. \](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
6 replies
BBNoDollar
May 18, 2025
BBNoDollar
an hour ago
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Hard Inequality
danilorj   5
N 2 hours ago by danilorj
Let $a, b, c > 0$ with $a + b + c = 1$. Prove that:
\[ \sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3}, \]with equality if and only if $a = b = c = \frac{1}{3}$.
5 replies
danilorj
Today at 5:17 AM
danilorj
2 hours ago
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easy geo
ErTeeEs06   5
N 2 hours ago by Adywastaken
Source: BxMO 2025 P3
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.
5 replies
ErTeeEs06
Apr 26, 2025
Adywastaken
2 hours ago
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Inspired by SXJX (12)2022 Q1167
sqing   4
N 2 hours ago by sqing
Source: Own
Let $ a,b,c>0 $. Prove that$$\frac{kabc-1} {abc(a+b+c+8(2k-1))}\leq \frac{1}{16 }$$Where $ k>\frac{1}{2}.$
4 replies
sqing
Yesterday at 4:01 AM
sqing
2 hours ago
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Geometry hard problem.
noneofyou34   3
N 2 hours ago by noneofyou34
In a circle of radius R, three chords of length R are given. Their ends are joined with segments to
obtain a hexagon inscribed in the circle. Show that the midpoints of the new chords are the vertices of
an equilateral triang
3 replies
noneofyou34
Today at 9:50 AM
noneofyou34
2 hours ago
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Where are the Circles?
luminescent   45
N 2 hours ago by math-olympiad-clown
Source: EGMO 2022/1
Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.
45 replies
luminescent
Apr 9, 2022
math-olympiad-clown
2 hours ago
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Find the minimum
sqing   9
N 2 hours ago by sqing
Source: China Shandong High School Mathematics Competition 2025 Q4
Let $ a,b,c>0,abc>1$. Find the minimum value of $ \frac {abc(a+b+c+8)}{abc-1}. $
9 replies
1 viewing
sqing
May 17, 2025
sqing
2 hours ago
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Functionnal equation
Rayanelba   1
N Apr 30, 2025 by Mrcuberoot
Source: Own
Find all functions $f:\mathbb{R}_{>0}\to \mathbb{R}_{>0}$ that verify the following equation for all $x,y\in \mathbb{R}_{>0}$:
$f(x+yf(x))+f(\frac{x}{y})=\frac{x}{y}+f(x+xy)$
1 reply
Rayanelba
Apr 30, 2025
Mrcuberoot
Apr 30, 2025
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Functionnal equation
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Reveal topic tags
algebra
functional equation
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Source: Own
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Rayanelba
20 posts
Rayanelba
#1 h  Apr 30, 2025, 4:10 PM
Y by
Find all functions $f:\mathbb{R}_{>0}\to \mathbb{R}_{>0}$ that verify the following equation for all $x,y\in \mathbb{R}_{>0}$:
$f(x+yf(x))+f(\frac{x}{y})=\frac{x}{y}+f(x+xy)$
This post has been edited 2 times. Last edited by Rayanelba, Apr 30, 2025, 7:25 PM
Reason: Typo
Z K Y
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This post has been deleted. Click here to see post.
Mrcuberoot
28 posts
Mrcuberoot
#2 h  Apr 30, 2025, 6:27 PM • 2 Y
Y by Rayanelba, ATM_
Unsolvable probably
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