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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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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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$7^{7^n}+1$ is the product of at least $2n + 3$ primes
N.T.TUAN   46
N 29 minutes ago by reni_wee
Source: USAMO 2007
Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.
46 replies
N.T.TUAN
Apr 26, 2007
reni_wee
29 minutes ago
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n x n square and strawberries
pohoatza   19
N 33 minutes ago by atdaotlohbh
Source: IMO Shortlist 2006, Combinatorics 4, AIMO 2007, TST 4, P2
A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$.

Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows:

A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.
19 replies
pohoatza
Jun 28, 2007
atdaotlohbh
33 minutes ago
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Consecutive squares are floors
ICE_CNME_4   7
N 43 minutes ago by ICE_CNME_4

Determine how many positive integers \( n \) have the property that both
\[ \left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor \]are consecutive perfect squares.
7 replies
ICE_CNME_4
Today at 1:50 PM
ICE_CNME_4
43 minutes ago
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2024 Miklós-Schweitzer problem 3
Martin.s   2
N an hour ago by NODIRKHON_UZ
Do there exist continuous functions $f, g: \mathbb{R} \to \mathbb{R}$, both nowhere differentiable, such that $f \circ g$ is differentiable?
2 replies
Martin.s
Dec 5, 2024
NODIRKHON_UZ
an hour ago
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Hard geo finale with the cursed line
hakN   12
N an hour ago by ihategeo_1969
Source: 2024 Turkey TST P9
In a scalene triangle $ABC,$ $I$ is the incenter and $O$ is the circumcenter. The line $IO$ intersects the lines $BC,CA,AB$ at points $D,E,F$ respectively. Let $A_1$ be the intersection of $BE$ and $CF$. The points $B_1$ and $C_1$ are defined similarly. The incircle of $ABC$ is tangent to sides $BC,CA,AB$ at points $X,Y,Z$ respectively. Let the lines $XA_1, YB_1$ and $ZC_1$ intersect $IO$ at points $A_2,B_2,C_2$ respectively. Prove that the circles with diameters $AA_2,BB_2$ and $CC_2$ have a common point.
12 replies
hakN
Mar 18, 2024
ihategeo_1969
an hour ago
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two lines passsing through the midpoint
miiirz30   1
N an hour ago by optimusprime154
Source: 2025 Euler Olympiad, Round 2
Points $A$, $B$, $C$, and $D$ lie on a line in that order, and points $E$ and $F$ are located outside the line such that $EA=EB$, $FC=FD$ and $EF \parallel AD$. Let the circumcircles of triangles $ABF$ and $CDE$ intersect at points $P$ and $Q$, and the circumcircles of triangles $ACF$ and $BDE$ intersect at points $M$ and $N$. Prove that the lines $PQ$ and $MN$ pass through the midpoint of segment $EF$.

Proposed by Giorgi Arabidze, Georgia
1 reply
miiirz30
Today at 10:23 AM
optimusprime154
an hour ago
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2024 Miklós Schweitzer problem 2
Martin.s   1
N an hour ago by NODIRKHON_UZ
Does there exist a nowhere dense, nonempty compact set $C \subset [0,1]$ such that
\[ \liminf_{h \to 0^+} \frac{\lambda(C \cap (x, x+h))}{h} > 0 \quad \text{or} \quad \liminf_{h \to 0^+} \frac{\lambda(C \cap (x-h, x))}{h} > 0 \]holds for every point $x \in C$, where $\lambda(A)$ denotes the Lebesgue measure of $A$?
1 reply
Martin.s
Dec 5, 2024
NODIRKHON_UZ
an hour ago
J
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Upper bound on products in sequence
tapir1729   11
N an hour ago by HamstPan38825
Source: TSTST 2024, problem 7
An infinite sequence $a_1$, $a_2$, $a_3$, $\ldots$ of real numbers satisfies
\[ a_{2n-1} + a_{2n} > a_{2n+1} + a_{2n+2} \qquad \mbox{and} \qquad a_{2n} + a_{2n+1} < a_{2n+2} + a_{2n+3} \]for every positive integer $n$. Prove that there exists a real number $C$ such that $a_{n} a_{n+1} < C$ for every positive integer $n$.

Merlijn Staps
11 replies
tapir1729
Jun 24, 2024
HamstPan38825
an hour ago
J
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Long and wacky inequality
Royal_mhyasd   4
N an hour ago by Royal_mhyasd
Source: Me
Let $x, y, z$ be positive real numbers such that $x^2 + y^2 + z^2 = 12$. Find the minimum value of the following sum :
$$\sum_{cyc}\frac{(x^3+2y)^3}{3x^2yz - 16z - 8yz + 6x^2z}$$knowing that the denominators are positive real numbers.
4 replies
1 viewing
Royal_mhyasd
May 12, 2025
Royal_mhyasd
an hour ago
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perpendicular diagonals criterion for a cyclic quadrilateral
parmenides51   3
N 2 hours ago by PEKKA
Source: Sharygin 2005 Finals 9.1
The quadrangle $ABCD$ is inscribed in a circle whose center $O$ lies inside it.
Prove that if $\angle BAO = \angle DAC$, then the diagonals of the quadrilateral are perpendicular.
3 replies
parmenides51
Aug 26, 2019
PEKKA
2 hours ago
J
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functional inequality with equality
miiirz30   3
N 2 hours ago by genius_007
Source: 2025 Euler Olympiad, Round 2
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that the following two conditions hold:

1. For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$.

2. For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$.

Proposed by Zaza Melikidze, Georgia
3 replies
miiirz30
Today at 10:32 AM
genius_007
2 hours ago
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JBMO Shortlist 2023 N6
Orestis_Lignos   4
N 2 hours ago by MR.1
Source: JBMO Shortlist 2023, N6
Version 1. Find all primes $p$ satisfying the following conditions:

(i) $\frac{p+1}{2}$ is a prime number.
(ii) There are at least three distinct positive integers $n$ for which $\frac{p^2+n}{p+n^2}$ is an integer.

Version 2. Let $p \neq 5$ be a prime number such that $\frac{p+1}{2}$ is also a prime. Suppose there exist positive integers $a <b$ such that $\frac{p^2+a}{p+a^2}$ and $\frac{p^2+b}{p+b^2}$ are integers. Show that $b=(a-1)^2+1$.
4 replies
Orestis_Lignos
Jun 28, 2024
MR.1
2 hours ago
J
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External Direct Sum
We2592   1
N 5 hours ago by Acridian9
Q) 1. Let $V$ be external direct sum of vector spaces $U$ and $W$ over a field $\mathbb{F}$.let $\hat{U}={\{(u,0):u\in U\}}$ and $\hat{W}={\{(0,w):w\in W\}}$
show that
i) $\hat{U}$ and $\hat{W}$ is subspaces.
ii)$V=\hat{U}\oplus\hat{W}$

Q)2. Suppose $V=U+W$. Let $\hat{V}$ be the external direct sum of $U$ and $W$. show that $V$ is isomorphic to $\hat{V}$ under the correspondence $v=u+w\leftrightarrow(u,w)$

I face some trouble to solve this problems help me for understanding.
thank you.

1 reply
We2592
Yesterday at 2:45 AM
Acridian9
5 hours ago
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How can I know the sequences's convergence value?
Madunglecha   5
N Today at 10:50 AM by teomihai
What is the convergence value of the sequence??
(n^2)*ln(n+1/n)-n
5 replies
Madunglecha
Yesterday at 6:56 AM
teomihai
Today at 10:50 AM
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J
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Rank and block matrix
KAME06   4
N Apr 17, 2025 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
Apr 17, 2025
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Rank and block matrix
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Reveal topic tags
linear algebra
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KAME06
160 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, Apr 17, 2025, 4:23 PM
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c00lb0y
14 posts
c00lb0y
#2 h  Apr 17, 2025, 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
14 posts
c00lb0y
#3 h  Apr 17, 2025, 10:09 AM
Y by
sorry for such bad writing, i needda learn latex or something like that
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loup blanc
3599 posts
loup blanc
#4 h  Apr 17, 2025, 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
160 posts
KAME06
#5 h  Apr 17, 2025, 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, Apr 17, 2025, 4:20 PM
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