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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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Convergent series with weight becomes divergent
P_Fazioli   0
10 minutes ago
Initially, my problem was : is it true that if we fix $(b_n)$ positive such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$, then there exists $(a_n)$ positive such that $\displaystyle\sum_{n\geq 0}a_n$ converges and $\displaystyle\sum_{n\geq 0}a_nb_n$ diverges ?

Thinking about the continuous case : if $g:\mathbb{R}_+\longrightarrow\mathbb{R}$ is continuous, positive with $g(x)\underset{{x}\longrightarrow{+\infty}}\longrightarrow +\infty$, does $f$ continuous and positive exist on $\mathbb{R}_+$ such that $\displaystyle\int_0^{+\infty}f(x)\text{d}x$ converges and $\displaystyle\int_0^{+\infty}f(x)g(x)\text{d}x$ diverges ?

To the last question, the answer seems to be yes if $g$ is in the $\mathcal{C}^1$ class, increasing : I chose $f=\dfrac{g'}{g^2}$. With this idea, I had the idea to define $a_n=\dfrac{b_{n+1}-b_n}{b_n^2}$ but it is not clear that it is ok, even if $(b_n)$ is increasing.

Now I have some questions !

1) The main problem : is it true that if we fix $(b_n)$ positive such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$, then there exists $(a_n)$ positive such that $\displaystyle\sum_{n\geq 0}a_n$ converges and $\displaystyle\sum_{n\geq 0}a_nb_n$ diverges ? And if $(b_n)$ is increasing ?
2) is it true that if we fix $(b_n)$ positive increasing such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}\underset{{n}\longrightarrow{+\infty}}\longrightarrow 1$, then $\displaystyle\sum_{n\geq 0}\left(\frac{b_{n+1}}{b_n}-1\right)$ diverges ?
3) is it true that if we fix $(b_n)$ positive increasing such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}\underset{{n}\longrightarrow{+\infty}}\longrightarrow 1$, then $\displaystyle\sum_{n\geq 0}\frac{b_{n+1}-b_n}{b_n^2}$ converges ?
4) if $(b_n)$ is positive increasing and such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}$ does not converge to $1$, can $\displaystyle\sum_{n\geq 0}\frac{b_{n+1}-b_n}{b_n^2}$ diverge ?
5) for the continuous case, is it true if we suppose $g$ only to be continuous ?

0 replies
P_Fazioli
10 minutes ago
0 replies
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D1026 : An equivalent
Dattier   4
N an hour ago by 3ch03s
Source: les dattes à Dattier
Let $u_0=1$ and $\forall n \in \mathbb N, u_{2n+1}=\ln(1+u_{2n}), u_{2n+2}=\sin(u_{2n+1})$.

Find an equivalent of $u_n$.
4 replies
Dattier
Saturday at 1:39 PM
3ch03s
an hour ago
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How many numbers
brokendiamond   0
2 hours ago
How many 5-digit numbers can be formed using the digits 1, 3, 5, 7, 9 such that the smaller digits are not positioned between two larger digits?
0 replies
brokendiamond
2 hours ago
0 replies
J
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Sum of squares in 1865
Twoisaprime   2
N 2 hours ago by EmptyMachine
Source: 2024 CWMO P1
For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$.
Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$
2 replies
Twoisaprime
Aug 6, 2024
EmptyMachine
2 hours ago
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USA 97 [1/(b^3+c^3+abc) + ... >= 1/(abc)]
Maverick   47
N 2 hours ago by justaguy_69
Source: USAMO 1997/5; also: ineq E2.37 in Book: Inegalitati; Authors:L.Panaitopol,V. Bandila,M.Lascu
Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality
\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc} \]
holds.
47 replies
Maverick
Sep 12, 2003
justaguy_69
2 hours ago
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Inspired by old results
sqing   1
N 2 hours ago by sqing
Source: Own
Let $ a,b>0 $ and $ \frac{a}{b^2}+\frac{2\sqrt{3}}{2a^2+b^2}\leq 3\sqrt 3. $ Prove that$$a^2+2b^2\geq 1$$
1 reply
sqing
2 hours ago
sqing
2 hours ago
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Orthocenter
jayme   7
N 2 hours ago by Ianis
Dear Mathlinkers,

1. ABC an acuatangle triangle
2. H the orthcenter of ABC
3. DEF the orthic triangle of ABC
4. A* the midpoint of AH
5. X the point of intersection of AH and EF.

Prove : X is the orthocenter of A*BC.

Sincerely
Jean-Louis
7 replies
jayme
Mar 25, 2015
Ianis
2 hours ago
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something...
SunnyEvan   1
N 2 hours ago by SunnyEvan
Source: unknown
Try to prove : $$ \sum csc^{20} \frac{2^{i} \pi}{7} csc^{23} \frac{2^{j}\pi }{7} csc^{2023} \frac{2^{k} \pi}{7} $$is a rational number.
Where $ (i,j,k)=(1,2,3) $ and other permutations.
1 reply
SunnyEvan
5 hours ago
SunnyEvan
2 hours ago
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HK bisect QS
lssl   24
N 3 hours ago by LeYohan
Source: 1998 HK
In a concyclic quadrilateral $PQRS$,$\angle PSR=\frac{\pi}{2}$ , $H,K$ are perpendicular foot from $Q$ to sides $PR,RS$ , prove that $HK$ bisect segment$SQ$.
24 replies
lssl
Jan 5, 2012
LeYohan
3 hours ago
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Points in general position
AshAuktober   3
N 4 hours ago by blackbluecar
Source: 2025 Nepal ptst p1 of 4
Shining tells Prajit a positive integer $n \ge 2025$. Prajit then tries to place n points such that no four points are concyclic and no $3$ points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?

(Prajit Adhikari, Nepal and Shining Sun, USA)
3 replies
AshAuktober
Mar 15, 2025
blackbluecar
4 hours ago
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Interesting inequalities
sqing   2
N 4 hours ago by sqing
Source: Own
Let $ a,b,c>0 $ and $ a+b\leq 16abc. $ Prove that
$$ a+b+kc\geq\sqrt{k}$$$$ a+b+kc^2\geq\frac{3\sqrt[3]{k}}{4}$$Where $ k>0. $
$$ a+b+c\geq1$$$$ a+b+4c\geq2$$$$ a+b+c^2\geq\frac{3}{4}$$$$ a+b+8c^2\geq\frac{3}{2}$$
2 replies
sqing
Yesterday at 12:23 PM
sqing
4 hours ago
J
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IMO 2014 Problem 1
Amir Hossein   132
N 4 hours ago by maromex
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
Proposed by Gerhard Wöginger, Austria.
132 replies
Amir Hossein
Jul 8, 2014
maromex
4 hours ago
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Rolles theorem
sasu1ke   4
N Yesterday at 6:13 PM by sasu1ke

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that
\[ f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1. \]Prove that there exists a point \( \xi \in (0, 1) \) such that
\[ f(\xi) \cdot f'(\xi) + f''(\xi) = 0. \]

4 replies
sasu1ke
Saturday at 9:00 PM
sasu1ke
Yesterday at 6:13 PM
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Putnam 1947 B1
sqrtX   3
N Yesterday at 5:38 PM by Levieee
Source: Putnam 1947
Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$
$$f'(x)= \frac{1}{x^2 +f(x)^{2}}.$$Prove that
$$\lim_{x\to \infty} f(x)$$exists and is less than $1+ \frac{\pi}{4}.$
3 replies
sqrtX
Apr 3, 2022
Levieee
Yesterday at 5:38 PM
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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
matrix
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KAME06
158 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
6 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
6 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
3591 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
158 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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