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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
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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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My Unsolved Problem
ZeltaQN2008   0
13 minutes ago
Source: IDK
Given a positive integer \( m \) and \( a > 1 \). Prove that there always exists a positive integer \( n \) such that \( m \mid (a^n + n) \).

P/s: I can prove the problem if $m$ is a power of a prime number, but for arbitrary $m$ then well.....
0 replies
ZeltaQN2008
13 minutes ago
0 replies
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Computing functions
BBNoDollar   3
N 30 minutes ago by wh0nix
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 $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
3 replies
BBNoDollar
May 21, 2025
wh0nix
30 minutes ago
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Computing functions
BBNoDollar   8
N 31 minutes ago by wh0nix
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 $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
8 replies
BBNoDollar
May 18, 2025
wh0nix
31 minutes ago
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Find the remainder
Jackson0423   1
N 33 minutes ago by wh0nix

Find the remainder when
\[ \frac{5^{2000} - 1}{4} \]is divided by \(64\).
1 reply
Jackson0423
2 hours ago
wh0nix
33 minutes ago
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IMO 2018 Problem 1
juckter   170
N 38 minutes ago by Adywastaken
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece
170 replies
juckter
Jul 9, 2018
Adywastaken
38 minutes ago
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Nice "if and only if" function problem
ICE_CNME_4   2
N 40 minutes ago by wh0nix
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 $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )

Please do it at 9th grade level. Thank you!
2 replies
ICE_CNME_4
Yesterday at 7:23 PM
wh0nix
40 minutes ago
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Minimum of this fuction
persamaankuadrat   1
N an hour ago by alexheinis
Source: KTOM January 2020
If $x$ is a positive real number, find the minimum of the following expression

$$\lfloor x \rfloor + \frac{500}{\lceil x\rceil^{2}}$$
1 reply
persamaankuadrat
3 hours ago
alexheinis
an hour ago
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A well-known circle hides in the dark
darij grinberg   15
N 2 hours ago by TUAN2k8
Source: All-Russian Olympiad 2006 finals, problem 11.4 (problem 4 for grade 11)
Given a triangle $ ABC$. The angle bisectors of the angles $ ABC$ and $ BCA$ intersect the sides $ CA$ and $ AB$ at the points $ B_1$ and $ C_1$, and intersect each other at the point $ I$. The line $ B_1C_1$ intersects the circumcircle of triangle $ ABC$ at the points $ M$ and $ N$. Prove that the circumradius of triangle $ MIN$ is twice as long as the circumradius of triangle $ ABC$.
15 replies
darij grinberg
May 6, 2006
TUAN2k8
2 hours ago
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unsolved nt
MuradSafarli   1
N 2 hours ago by whwlqkd
Find all prime numbers $p$ such that $p^2 - 12$ is also a prime number.
1 reply
MuradSafarli
2 hours ago
whwlqkd
2 hours ago
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IMO Problem 4
iandrei   106
N 2 hours ago by alexanderchew
Source: IMO ShortList 2003, geometry problem 1
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
106 replies
iandrei
Jul 14, 2003
alexanderchew
2 hours ago
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interesting diophantiic fe in natural numbers
skellyrah   0
3 hours ago
Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that for all \( m, n \in \mathbb{N} \),
\[ mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!). \]
0 replies
skellyrah
3 hours ago
0 replies
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Three numbers cannot be squares simultaneously
WakeUp   39
N 3 hours ago by MR.1
Source: APMO 2011
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.
39 replies
WakeUp
May 18, 2011
MR.1
3 hours ago
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FE over R
IAmTheHazard   21
N 3 hours ago by benjaminchew13
Source: ELMO Shortlist 2024/A3
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$
Andrew Carratu
21 replies
IAmTheHazard
Jun 22, 2024
benjaminchew13
3 hours ago
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Nice one
Blacklord   10
N 3 hours ago by Pal702004
Source: ....
Find all integers numbers (a,b,c) such that
$a/b + b/c + c/a =3$
10 replies
Blacklord
Jan 26, 2017
Pal702004
3 hours ago
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Number of complex solutions (x,y,z)
CarSa   1
N Apr 29, 2025 by Mathzeus1024
Find all solutions $(x,y,z)$ to the system of equations
\[\begin{aligned} \begin{cases} x^2+y^2-xy-3x+3=0,\\ x^2+y^2+z^2-xy-yz-2zx-3x+3=0.\\ \end{cases} \end{aligned}\]
1 reply
CarSa
Apr 27, 2025
Mathzeus1024
Apr 29, 2025
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Number of complex solutions (x,y,z)
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Reveal topic tags
algebra
system of equations
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CarSa
3 posts
CarSa
#1 h  Apr 27, 2025, 1:18 AM
Y by
Find all solutions $(x,y,z)$ to the system of equations
\[\begin{aligned} \begin{cases} x^2+y^2-xy-3x+3=0,\\ x^2+y^2+z^2-xy-yz-2zx-3x+3=0.\\ \end{cases} \end{aligned}\]
This post has been edited 1 time. Last edited by CarSa, Apr 27, 2025, 1:24 AM
Reason: the title was incorrect
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Mathzeus1024
919 posts
Mathzeus1024
#2 h  Apr 29, 2025, 10:35 AM
Y by
Let $x,y,z \in \mathbb{C}$ such that from the first equation:

$y = \frac{x \pm \sqrt{x^2-4(1)(x^2-3x+3)}}{2} = \frac{x \pm \sqrt{-3x^2+12x-12}}{2} = \frac{x \pm \sqrt{3}(x-2)i}{2}$;

or $x=t, y = \frac{t \pm \sqrt{3}(t-2)i}{2}$ for any $t \in \mathbb{C}$. Now, substitution of the first equation into the second gives us the reduced form:

$z^2-yz-2xz=0 \Rightarrow z[z-(y+2x)]=0 \Rightarrow z = 0, y+2x$;

which ultimately yields the triplets:

$\textcolor{red}{(x,y,z) = \left(t, \frac{t \pm \sqrt{3}(t-2)i}{2}, 0\right); \left(t, \frac{t \pm \sqrt{3}(t-2)i}{2}, \frac{5t \pm \sqrt{3}(t-2)i}{2}\right)}$.
This post has been edited 1 time. Last edited by Mathzeus1024, Apr 29, 2025, 10:35 AM
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