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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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Prove that 4p-3 is a square - Iran NMO 2005 - Problem1
sororak   24
N 34 minutes ago by EmersonSoriano
Let $n,p>1$ be positive integers and $p$ be prime. We know that $n|p-1$ and $p|n^3-1$. Prove that $4p-3$ is a perfect square.
24 replies
sororak
Sep 21, 2010
EmersonSoriano
34 minutes ago
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IMO ShortList 1999, combinatorics problem 4
orl   27
N 41 minutes ago by cursed_tangent1434
Source: IMO ShortList 1999, combinatorics problem 4
Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.
27 replies
orl
Nov 14, 2004
cursed_tangent1434
41 minutes ago
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Evan Chen Multivariable Calculus Book
Existing_Human1   0
an hour ago
What do you guys think of Evan Chen's multivariable calculus book (or whatever you want to call it), through MIT? Is it useful for learning multivariable calculus? Here is a link: book
0 replies
Existing_Human1
an hour ago
0 replies
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Geometry
Lukariman   2
N an hour ago by Lukariman
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that <HDM = 2∠AMP.
2 replies
Lukariman
Yesterday at 12:43 PM
Lukariman
an hour ago
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IMO 2010 Problem 1
canada   119
N an hour ago by lpieleanu
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$

Proposed by Pierre Bornsztein, France
119 replies
canada
Jul 7, 2010
lpieleanu
an hour ago
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f(n) <= f(a(G)) + f(b(G))
dangerousliri   5
N an hour ago by awesomeming327.
Source: FEOO, Problem 2, Shortlist C2
Given a group $G$ of people, we define $a(G)$ to be the least number of tables needed such that each person from the group sits in one of them with no two friends sitting on the same table, and $b(G)$ to be the least number of tables needed such that each person from the group sits in one of them with no two enemies sitting on the same table.
Consider all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each group $G$ of $n$ people we have
\[ f(n) \leqslant f(a(G)) + f(b(G)) \quad \text{and} \quad f(2) = 1.\]Find all possible values of $f(2020)$.

Note: We assume that for every pair of people in a group, they are either both friends with each other or both enemies with each other. Also, $\mathbb{N}$ is the set of all positive integers.

Proposed by Demetres Christofides, Cyprus
5 replies
dangerousliri
May 30, 2020
awesomeming327.
an hour ago
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f(w^2+x^2+y^2+z^2)=f(w^2+x^2)+f(y^2+z^2)
dangerousliri   23
N 2 hours ago by awesomeming327.
Source: FEOO, Problem 1, Shortlist N1
Find all functions $f:\mathbb{N}_0\rightarrow\mathbb{R}$ such that,
$$f(w^2+x^2+y^2+z^2)=f(w^2+x^2)+f(y^2+z^2)$$for all non-negative integers $w,x,y$ and $z$.

Note: $\mathbb{N}_0$ is the set of all non-negative integers and $\mathbb{R}$ is the set of all real numbers.

Proposed by Dorlir Ahmeti, Kosovo
23 replies
dangerousliri
May 30, 2020
awesomeming327.
2 hours ago
J
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Darboux cubic
srirampanchapakesan   0
2 hours ago
Source: Own
Let P be a point on the Darboux cubic (or the McCay Cubic ) of triangle ABC.

P1P2P3 is the circumcevian or pedal triangle of P wrt ABC.

Prove that P also lie on the Darboux cubic ( or the McCay Cubic) of P1P2P3 .
0 replies
srirampanchapakesan
2 hours ago
0 replies
J
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primitive polyominoes
N.T.TUAN   29
N 2 hours ago by Disjunction
Source: USAMO 2007
An animal with $n$ cells is a connected figure consisting of $n$ equal-sized cells[1].

A dinosaur is an animal with at least $2007$ cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.

(1) Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
29 replies
N.T.TUAN
Apr 26, 2007
Disjunction
2 hours ago
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Only consecutive terms are coprime
socrates   37
N 2 hours ago by blueprimes
Source: 7th RMM 2015, Problem 1
Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?
37 replies
socrates
Feb 28, 2015
blueprimes
2 hours ago
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Cyclic roots are not real, they can't hurt you
anantmudgal09   21
N 2 hours ago by bjump
Source: INMO 2023 P2
Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$:
$$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, i.e., $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real.

Proposed by Prithwijit De
21 replies
anantmudgal09
Jan 15, 2023
bjump
2 hours ago
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Rolles theorem
sasu1ke   7
N Yesterday at 7:27 PM by GentlePanda24

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
\[ f(\xi) \cdot f'(\xi) + f''(\xi) = 0. \]

7 replies
sasu1ke
May 3, 2025
GentlePanda24
Yesterday at 7:27 PM
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Morphism in a ring makes it a field
RobertRogo   0
Yesterday at 4:02 PM
Source: Daniel Jinga, Ionel Popescu, RNMO SHL, 2003
Let $A$ be a ring with unity in which $1+1 \neq 0$ and there is a morphism $f$ from the group $(A,+)$ to the monoid $(A,\cdot)$ such that for all $a\in A\setminus \{0\}$, there is a $b \in A$ such that $f(b)=a^2$. Prove that $A$ is a field.
0 replies
RobertRogo
Yesterday at 4:02 PM
0 replies
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Convergent series with weight becomes divergent
P_Fazioli   7
N Yesterday at 3:07 PM by P_Fazioli
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 ?

7 replies
P_Fazioli
Monday at 5:37 AM
P_Fazioli
Yesterday at 3:07 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...
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Samujjal101
2799 posts
Samujjal101
#2 h  Apr 28, 2025, 3:06 PM
Y by
What do you mean by dependent ?
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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
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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
1564 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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