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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 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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2025 OMOUS Problem 1
enter16180   1
N 21 minutes ago by KAME06
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Aman and Berdi, two biologists, they invented a new type of bacteria such that they can control the division of bacteria into several parts. They are also participants of $OMOUS-2025$ with the aim to train for the first problem of $OMOUS-2025$. They play the following game.
Initially, they take $1$ bacteria and choose a natural number $n$. On each move, the player chooses any $k$ number from $1$ to $n$. Then the player divides each bacterium into $k$ pants. Once chosen, the number $k$ cannot be chosen twice. If after any player's move the number of bacteria population is divisible by $n$ then that player loses. Determine who has the winning strategy depending on the given number $n$ if it's known that Amman starts first.
1 reply
enter16180
Today at 11:44 AM
KAME06
21 minutes ago
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Interesting Limit
Riptide1901   0
an hour ago
Find $\displaystyle\lim_{x\to\infty}\left|f(x)-\Gamma^{-1}(x)\right|$ where $\Gamma^{-1}(x)$ is the inverse gamma function, and $f^{-1}$ is the inverse of $f(x)=x^x.$
0 replies
Riptide1901
an hour ago
0 replies
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Bijection on the set of integers
talkon   19
N an hour ago by AN1729
Source: InfinityDots MO 2 Problem 2
Determine all bijections $f:\mathbb Z\to\mathbb Z$ satisfying
$$f^{f(m+n)}(mn) = f(m)f(n)$$for all integers $m,n$.

Note: $f^0(n)=n$, and for any positive integer $k$, $f^k(n)$ means $f$ applied $k$ times to $n$, and $f^{-k}(n)$ means $f^{-1}$ applied $k$ times to $n$.

Proposed by talkon
19 replies
talkon
Apr 9, 2018
AN1729
an hour ago
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Number Theory
TUAN2k8   1
N an hour ago by Roger.Moore
Find all positve integers m such that $m+1 | 3^m+1$
1 reply
TUAN2k8
4 hours ago
Roger.Moore
an hour ago
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D1020 : A strange result of number theory
Dattier   0
2 hours ago
Source: les dattes à Dattier
Let $x>1,n \in \mathbb N^*$ with $\gcd(E(x\times 10^n),E(x \times 10^{n+1}))=1 $ and $p=E(x\times 10^n)$ prime number.

Is it true that $\forall m \in\mathbb N,m>n, \gcd(p,E(10^m\times x))=1$?

PS : $E$ is the function integer part, hence $E(1.9)=1$.
0 replies
Dattier
2 hours ago
0 replies
J
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Continuity of function and line segment of integer length
egxa   1
N 2 hours ago by tonykuncheng
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 \( N \).
1 reply
egxa
2 hours ago
tonykuncheng
2 hours ago
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Polynomial x-axis angle
egxa   1
N 2 hours ago by Fishheadtailbody
Source: All Russian 2025 9.5
Let \( P_1(x) \) and \( P_2(x) \) be monic quadratic trinomials, and let \( A_1 \) and \( A_2 \) be the vertices of the parabolas \( y = P_1(x) \) and \( y = P_2(x) \), respectively. Let \( m(g(x)) \) denote the minimum value of the function \( g(x) \). It is known that the differences \( m(P_1(P_2(x))) - m(P_1(x)) \) and \( m(P_2(P_1(x))) - m(P_2(x)) \) are equal positive numbers. Find the angle between the line \( A_1A_2 \) and the $x$-axis.
1 reply
egxa
2 hours ago
Fishheadtailbody
2 hours ago
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Strategy game based modulo 3
egxa   1
N 2 hours ago by Euler8038
Source: All Russian 2025 9.7
The numbers \( 1, 2, 3, \ldots, 60 \) are written in a row in that exact order. Igor and Ruslan take turns inserting the signs \( +, -, \times \) between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by $3$, Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves?
1 reply
egxa
2 hours ago
Euler8038
2 hours ago
J
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Find the maximum value of x^3+2y
BarisKoyuncu   8
N 2 hours ago by Primeniyazidayi
Source: 2021 Turkey JBMO TST P4
Let $x,y,z$ be real numbers such that $$\left|\dfrac yz-xz\right|\leq 1\text{ and }\left|yz+\dfrac xz\right|\leq 1$$Find the maximum value of the expression $$x^3+2y$$
8 replies
BarisKoyuncu
May 23, 2021
Primeniyazidayi
2 hours ago
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Woaah a lot of external tangents
egxa   0
2 hours ago
Source: All Russian 2025 11.7
A quadrilateral \( ABCD \) with no parallel sides is inscribed in a circle \( \Omega \). Circles \( \omega_a, \omega_b, \omega_c, \omega_d \) are inscribed in triangles \( DAB, ABC, BCD, CDA \), respectively. Common external tangents are drawn between \( \omega_a \) and \( \omega_b \), \( \omega_b \) and \( \omega_c \), \( \omega_c \) and \( \omega_d \), and \( \omega_d \) and \( \omega_a \), not containing any sides of quadrilateral \( ABCD \). A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle \( \Gamma \). Prove that the lines joining the centers of \( \omega_a \) and \( \omega_c \), \( \omega_b \) and \( \omega_d \), and the centers of \( \Omega \) and \( \Gamma \) all intersect at one point.
0 replies
egxa
2 hours ago
0 replies
J
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Petya and vasya are playing with ones
egxa   0
2 hours ago
Source: All Russian 2025 11.6
$100$ ones are written in a circle. Petya and Vasya take turns making \( 10^{10} \) moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by $2$. Vasya chooses $10$ consecutive numbers and increases each by $1$. They alternate turns, starting with Petya. Prove that Vasya can act in such a way that after each of his moves, there are always at least five positive numbers, regardless of how Petya plays.
0 replies
egxa
2 hours ago
0 replies
J
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Outcome related combinatorics problem
egxa   0
2 hours ago
Source: All Russian 2025 10.7
A competition consists of $25$ sports, each awarding one gold medal to a winner. $25$ athletes participate, each in all $25$ sports. There are also $25$ experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to $25$. An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number \( k \) such that the experts can make their predictions so that at least \( k \) of them are guaranteed to be competent regardless of the outcome?
0 replies
egxa
2 hours ago
0 replies
J
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Sequence of functions
Squeeze   1
N 2 hours ago by Squeeze
Q) let $f_n:[-1,1)\to\mathbb{R}$ and $f_n(x)=x^{n}$ then is this uniformly convergence on $(0,1)$ comment on uniformly convergence on $[0,1]$ where in general it is should be uniformly convergence.

My I am trying with some contradicton method like chose $\epsilon=1$ and trying to solve$|f_n(a)-f(a)|<\epsilon=1$
Next take a in (0,1) and chose a= 2^1/N but not solution
How to solve like this way help.
1 reply
Squeeze
Today at 3:56 AM
Squeeze
2 hours ago
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Integrate lnx/sqrt{1-x^2}
EthanWYX2009   1
N 4 hours ago by GreenKeeper
Determine the value of
\[I=\int\limits_{0}^{1}\frac{\ln x}{\sqrt{1-x^2}}\mathrm dx.\]
1 reply
EthanWYX2009
5 hours ago
GreenKeeper
4 hours ago
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J
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Rank and block matrix
KAME06   4
N Yesterday at 4:19 PM by KAME06
Let $A, B, C, D$ matrices such $A$ is invertible.
Prove that if $Rank(A)=Rank(\begin{bmatrix} A & B \\ C & D \end{bmatrix}$), then $D=CA^{-1}B$
4 replies
KAME06
Apr 14, 2025
KAME06
Yesterday at 4:19 PM
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Rank and block matrix
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Reveal topic tags
linear algebra
matrix
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KAME06
148 posts
KAME06
#1 h  Apr 14, 2025, 7:46 PM
Y by
Let $A, B, C, D$ matrices such $A$ is invertible.
Prove that if $Rank(A)=Rank(\begin{bmatrix} A & B \\ C & D \end{bmatrix}$), then $D=CA^{-1}B$
This post has been edited 2 times. Last edited by KAME06, Yesterday at 4:23 PM
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c00lb0y
6 posts
c00lb0y
#2 h  Yesterday at 10:08 AM • 1 Y
Y by KAME06
if range equals then rank of these matrices are also equal, then in rank we can do column, row operations that doesn't change rank.
rankA=rank/size]A B-A*A^(-1)B , C D-CA^(-1)B[size=200 so rankA>=rankA+rank(D-CA^(-1)B) , hence D=CA^(-1)B
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c00lb0y
6 posts
c00lb0y
#3 h  Yesterday at 10:09 AM
Y by
sorry for such bad writing, i needda learn latex or something like that
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loup blanc
3576 posts
loup blanc
#4 h  Yesterday at 2:13 PM • 2 Y
Y by KAME06, c00lb0y
@KAME06 , the range is the image. Here the correct word is rank.
c00lb0y's proof.
$rank(A)=rank(\begin{pmatrix} A & B \\ C & D \end{pmatrix})=rank(\begin{pmatrix} A & B \\ C & D \end{pmatrix}-\begin{pmatrix}\begin{pmatrix}0\\0\end{pmatrix}&\begin{pmatrix}A\\C\end{pmatrix}A^{-1}B\end{pmatrix})$ (combination of columns)
$=rank(\begin{pmatrix}A&0\\C&D-CA^{-1}B\end{pmatrix})\geq rank(A)+rank(D-CA^{-1}B)$ and $D=CA^{-1}B$. $\square$
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KAME06
148 posts
KAME06
#5 h  Yesterday at 4:19 PM
Y by
loup blanc wrote:
@KAME06 , the range is the image. Here the correct word is rank.
c00lb0y's proof.
$rank(A)=rank(\begin{pmatrix} A & B \\ C & D \end{pmatrix})=rank(\begin{pmatrix} A & B \\ C & D \end{pmatrix}-\begin{pmatrix}\begin{pmatrix}0\\0\end{pmatrix}&\begin{pmatrix}A\\C\end{pmatrix}A^{-1}B\end{pmatrix})$ (combination of columns)
$=rank(\begin{pmatrix}A&0\\C&D-CA^{-1}B\end{pmatrix})\geq rank(A)+rank(D-CA^{-1}B)$ and $D=CA^{-1}B$. $\square$

Oh, thanks, I'll fix it
This post has been edited 1 time. Last edited by KAME06, Yesterday at 4:20 PM
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