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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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Random walk
EthanWYX2009   0
29 minutes ago
As shown in the graph, an ant starts from $4$ and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching $1$ without passing through $6.$
0 replies
EthanWYX2009
29 minutes ago
0 replies
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Lemma on tangency involving a parallelogram with orthocenter
Gimbrint   0
35 minutes ago
Source: Own
Let $ABC$ be an acute triangle ($AB<BC$) with circumcircle $\omega$ and orthocenter $H$. Let $M$ be the midpoint of $AC$. Line $BH$ intersects $\omega$ again at $L\neq B$, and line $ML$ intersects $\omega$ again at $P\neq L$. Points $D$ and $E$ lie on $AB$ and $BC$ respectively, such that $BEHD$ is a parallelogram.

Prove that $BP$ is tangent to the circumcircle of triangle $BDE$.
0 replies
+1 w
Gimbrint
35 minutes ago
0 replies
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Consecutive squares are floors
ICE_CNME_4   10
N 40 minutes ago by JARP091

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.
10 replies
ICE_CNME_4
Yesterday at 1:50 PM
JARP091
40 minutes ago
J
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Sharygin 2025 CR P10
Gengar_in_Galar   2
N an hour ago by Kappa_Beta_725
Source: Sharygin 2025
An acute-angled triangle with one side equal to the altitude from the opposite vertex is cut from paper. Construct a point inside this triangle such that the square of the distance from it to one of the vertices equals the sum of the squares of distances to to the remaining two vertices. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines.
Proposed by: M.Evdokimov
2 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
an hour ago
J
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inequality thing
BinariouslyRandom   1
N an hour ago by lbh_qys
Source: Philippine MO 2025 P5
Find the largest real constant $k$ for which the inequality \[ (a^2+3)(b^2+3)(c^2+3)(d^2+3) + k(a-1)(b-1)(c-1)(d-1) \ge 0 \]holds for all real numbers $a$, $b$, $c$, and $d$.

answer
1 reply
BinariouslyRandom
an hour ago
lbh_qys
an hour ago
J
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Sharygin 2025 CR P12
Gengar_in_Galar   8
N 2 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Circles $\omega_{1}$ and $\omega_{2}$ are given. Let $M$ be the midpoint of the segment joining their centers, $X$, $Y$ be arbitrary points on $\omega_{1}$, $\omega_{2}$ respectively such that $MX=MY$. Find the locus of the midpoints of segments $XY$.
Proposed by: L Shatunov
8 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
2 hours ago
J
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Sharygin 2025 CR P17
Gengar_in_Galar   6
N 2 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Let $O$, $I$ be the circumcenter and the incenter of an acute-angled scalene triangle $ABC$; $D$, $E$, $F$ be the touching points of its excircle with the side $BC$ and the extensions of $AC$, $AB$ respectively. Prove that if the orthocenter of the triangle $DEF$ lies on the circumcircle of $ABC$, then it is symmetric to the midpoint of the arc $BC$ with respect to $OI$.
Proposed by: P.Puchkov,E.Utkin
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
2 hours ago
J
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Sharygin 2025 CR P21
Gengar_in_Galar   4
N 2 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur.
Proposed by: G.Galyapin
4 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
2 hours ago
J
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Sharygin 2025 CR P18
Gengar_in_Galar   6
N 2 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right
Proposed by: I.Kukharchuk
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
2 hours ago
J
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Problem 3
EthanWYX2009   5
N 2 hours ago by parkjungmin
Source: 2023 China Second Round P3
Find the smallest positive integer ${k}$ with the following properties $:{}{}{}{}{}$If each positive integer is arbitrarily colored red or blue${}{}{},$
there may be ${}{}{}{}9$ distinct red positive integers $x_1,x_2,\cdots ,x_9,$ satisfying
$$x_1+x_2+\cdots +x_8<x_9,$$or there are $10{}{}{}{}{}{}$ distinct blue positive integers $y_1,y_2,\cdots ,y_{10}$ satisfiying
$${y_1+y_2+\cdots +y_9<y_{10}}.$$
5 replies
EthanWYX2009
Sep 10, 2023
parkjungmin
2 hours ago
J
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Convergence of complex sequence
Rohit-2006   9
N Today at 3:41 AM by Saucitom
Suppose $z_1, z_2,\cdots,z_k$ are complex numbers with absolute value $1$. For $n=1,2,\cdots$ define $w_n=z_1^n+z_2^n+\cdots+z_k^n$. Given that the sequence $(w_n)_{n\geq1}$ converges. Show that,
$$z_1=z_2=\cdots=z_k=1$$.
9 replies
Rohit-2006
May 17, 2025
Saucitom
Today at 3:41 AM
J
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Invertible Matrices
Mateescu Constantin   8
N Yesterday at 8:44 PM by loup blanc
Source: Romanian District Olympiad 2018 - Grade XI - Problem 1
Show that if $n\ge 2$ is an integer, then there exist invertible matrices $A_1, A_2, \ldots, A_n \in \mathcal{M}_2(\mathbb{R})$ with non-zero entries such that:

\[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\]
Edit.
8 replies
Mateescu Constantin
Mar 10, 2018
loup blanc
Yesterday at 8:44 PM
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2024 Miklós-Schweitzer problem 3
Martin.s   2
N Yesterday at 7:01 PM 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
Yesterday at 7:01 PM
J
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2024 Mikl&oacute;s Schweitzer problem 2
Martin.s   1
N Yesterday at 6:43 PM 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
Yesterday at 6:43 PM
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Matrix Row and column relation.
Schro   6
N May 1, 2025 by Schro
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
6 replies
Schro
Apr 28, 2025
Schro
May 1, 2025
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Matrix Row and column relation.
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Reveal topic tags
linear algebra
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Schro
4 posts
Schro
#1 h  Apr 28, 2025, 2:54 PM
Y by
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
Z K Y
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Samujjal101
2801 posts
Samujjal101
#2 h  Apr 28, 2025, 3:06 PM
Y by
What do you mean by dependent ?
Z K Y
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rchokler
2975 posts
rchokler
#3 h  Apr 28, 2025, 8:40 PM
Y by
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.
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Schro
4 posts
Schro
#4 h  Apr 29, 2025, 4:31 PM
Y by
Samujjal101 wrote:
What do you mean by dependent ?


I mean if ith row is a combination of other rows.....then ith column will also be a combination of other columns
Z K Y
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Schro
4 posts
Schro
#5 h  Apr 29, 2025, 4:32 PM
Y by
rchokler wrote:
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.

I m not talking in general sense....
I m being specific.....if "i th" row is a combination of other rows.....then "i th " column will be a combination of other columns and vice versa
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Etkan
1570 posts
Etkan
#6 h  Apr 29, 2025, 5:01 PM
Y by
Schro wrote:
rchokler wrote:
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.

I m not talking in general sense....
I m being specific.....if "i th" row is a combination of other rows.....then "i th " column will be a combination of other columns and vice versa

It's not true. For example, in the matrix$$\begin{pmatrix}0&0&0\\1&0&0\\-1&0&0\end{pmatrix}$$the first row is the sum of the second and third rows, but any linear combination of the second and third columns is $0$, so it cannot be equal to the first column.
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Schro
4 posts
Schro
#7 h  May 1, 2025, 6:20 AM
Y by
Thnks, i got it
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