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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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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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2019 Iberoamerican Mathematical Olympiad, P1
jbaca   9
N an hour ago by jordiejoh
For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.
9 replies
jbaca
Sep 15, 2019
jordiejoh
an hour ago
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Conditional geo with centroid
a_507_bc   7
N an hour ago by Tkn
Source: Singapore Open MO Round 2 2023 P1
In a scalene triangle $ABC$ with centroid $G$ and circumcircle $\omega$ centred at $O$, the extension of $AG$ meets $\omega$ at $M$; lines $AB$ and $CM$ intersect at $P$; and lines $AC$ and $BM$ intersect at $Q$. Suppose the circumcentre $S$ of the triangle $APQ$ lies on $\omega$ and $A, O, S$ are collinear. Prove that $\angle AGO = 90^{o}$.
7 replies
a_507_bc
Jul 1, 2023
Tkn
an hour ago
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People live in Kansas?
jj_ca888   13
N an hour ago by Ilikeminecraft
Source: SMO 2020/5
In triangle $\triangle ABC$, let $E$ and $F$ be points on sides $AC$ and $AB$, respectively, such that $BFEC$ is cyclic. Let lines $BE$ and $CF$ intersect at point $P$, and $M$ and $N$ be the midpoints of $\overline{BF}$ and $\overline{CE}$, respectively. If $U$ is the foot of the perpendicular from $P$ to $BC$, and the circumcircles of triangles $\triangle BMU$ and $\triangle CNU$ intersect at second point $V$ different from $U$, prove that $A, P,$ and $V$ are collinear.

Proposed by Andrew Wen and William Yue
13 replies
jj_ca888
Aug 28, 2020
Ilikeminecraft
an hour ago
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Symmetric integer FE
a_507_bc   5
N an hour ago by Tkn
Source: Singapore Open MO Round 2 2023 P4
Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$for all integers $x, y$.
5 replies
a_507_bc
Jul 1, 2023
Tkn
an hour ago
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Channel name changed
Plane_geometry_youtuber   6
N an hour ago by Yiyj
Hi,

Due to the search handle issue in youtube. My channel is renamed to Olympiad Geometry Club. And the new link is as following:

https://www.youtube.com/@OlympiadGeometryClub

Recently I introduced the concept of harmonic bundle. I will move on to the conjugate median soon. In the future, I will discuss more than a thousand theorems on plane geometry and hopefully it can help to the students preparing for the Olympiad competition.

Please share this to the people may need it.

Thank you!
6 replies
Plane_geometry_youtuber
Yesterday at 9:31 PM
Yiyj
an hour ago
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How many cases did you check?
avisioner   18
N an hour ago by ezpotd
Source: 2023 ISL N2
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.

Proposed by Tahjib Hossain Khan, Bangladesh
18 replies
avisioner
Jul 17, 2024
ezpotd
an hour ago
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Beautiful geo but i cant solve this
phonghatemath   3
N an hour ago by phonghatemath
Source: homework
Given triangle $ABC$ inscribed in $(O)$. Two points $D, E$ lie on $BC$ such that $AD, AE$ are isogonal in $\widehat{BAC}$. $M$ is the midpoint of $AE$. $K$ lies on $DM$ such that $OK \bot AE$. $AD$ intersects $(O)$ at $P$. Prove that the line through $K$ parallel to $OP$ passes through the Euler center of triangle $ABC$.

Sorry for my English!
3 replies
phonghatemath
Yesterday at 4:48 PM
phonghatemath
an hour ago
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ISI UGB 2025
Entrepreneur   3
N an hour ago by Hello_Kitty
Source: ISI UGB 2025
1.)
Suppose $f:\mathbb R\to\mathbb R$ is differentiable and $|f'(x)|<\frac 12\;\forall\;x\in\mathbb R.$ Show that for some $x_0\in\mathbb R,f(x_0)=x_0.$

3.)
Suppose $f:[0,1]\to\mathbb R$ is differentiable with $f(0)=0.$ If $|f'(x)|\le f(x)\;\forall\;x\in[0,1],$ then show that $f(x)=0\;\forall\;x.$

4.)
Let $S^1=\{z\in\mathbb C:|z|=1\}$ be the unit circle in the complex plane. Let $f:S^1\to S^1$ be the map given by $f(z)=z^2.$ We define $f^{(1)}:=f$ and $f^{(k+1)}=f\circ f^{(k)}$ for $k\ge 1.$ The smallest positive integer $n$ such that $f^n(z)=z$ is called period of $z.$ Determine the total number of points $S^1$ of period $2025.$

6.)
Let $\mathbb N$ denote the set of natural numbers, and let $(a_i,b_i), 1\le i\le 9,$ be nine distinct tuples in $\mathbb N\times\mathbb N.$ Show that there are $3$ distinct elements in the set $\{2^{a_i}3^{b_i}:1\le i\le 9\}$ whose product is a perfect cube.

8.)
Let $n\ge 2$ and let $a_1\le a_2\le\cdots\le a_n$ be positive integers such that $$\sum_{i=1}^n a_i=\prod_{i=1}^n a_i.$$Prove that $$\sum_{i=1}^n a_i\le 2n$$and determine when equality holds.
3 replies
Entrepreneur
May 27, 2025
Hello_Kitty
an hour ago
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Product of Sum
shobber   5
N an hour ago by alexanderchew
Source: CGMO 2006
Given that $x_{i}>0$, $i = 1, 2, \cdots, n$, $k \geq 1$. Show that: \[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]
5 replies
shobber
Aug 9, 2006
alexanderchew
an hour ago
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Problem 12
SlovEcience   0
2 hours ago
Find all functions \( f: \mathbb{N} \to \mathbb{N} \) such that
\[ f(x^4 + 5y^4 + 10z^4) = f(x)^4 + 5f(y)^4 + 10f(z)^4 \]for all \( x, y, z \in \mathbb{N} \).
0 replies
SlovEcience
2 hours ago
0 replies
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Polynomial functional equation
Fishheadtailbody   3
N 2 hours ago by luutrongphuc
Source: MACMO
$P(x)$ is a polynomial with real coefficients such that
\[ P(x)^2 - 1 = 4 P(x^2 - 4x + 1). \]Find $P(x)$.

fixed now
3 replies
Fishheadtailbody
Apr 18, 2025
luutrongphuc
2 hours ago
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Expand into a Fourier series
Tip_pay   1
N 4 hours ago by maths001Z
Expand the function in a Fourier series on the interval $(-\pi, \pi)$
$$f(x)=\begin{cases} 1, & -1<x\leq 0\\ x, & 0<x<1 \end{cases}$$
1 reply
Tip_pay
Dec 12, 2023
maths001Z
4 hours ago
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functional analysis
ILOVEMYFAMILY   0
4 hours ago
Let \( E, F \) be normed vector spaces, where \( E \) is a Banach space, and let \( A_n \in \mathcal{L}(E, F) \).
Prove that the set
\[ X = \left\{ x \in E : \sup_{n \geq 1} \|A_n x\| < +\infty \right\} \]is either an empty set or second category.
0 replies
ILOVEMYFAMILY
4 hours ago
0 replies
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Prove that for every \( k \), there are infinitely many values of \( n \) such t
Martin.s   0
Yesterday at 7:07 PM
It is well known that
\[ \frac{(2n)!}{n! \cdot (n+1)!} \]is always an integer. Prove that for every \( k \), there are infinitely many values of \( n \) such that
\[ \frac{(2n)!}{n! \cdot (n+k)!} \]is an integer.
0 replies
Martin.s
Yesterday at 7:07 PM
0 replies
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COS(Matrix)
FFA21   1
N Apr 12, 2025 by loup blanc
Source: My head
1)Let $A\in M_{2\times 2}$ what is range of values of $cos(A)$
$cos(A)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}X^{2n}$
2) Let $A\in M_{n\times n}$ what is range of values of $cos(A)$
1 reply
FFA21
Apr 10, 2025
loup blanc
Apr 12, 2025
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COS(Matrix)
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Reveal topic tags
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FFA21
85 posts
FFA21
#1 h  Apr 10, 2025, 6:20 PM
Y by
1)Let $A\in M_{2\times 2}$ what is range of values of $cos(A)$
$cos(A)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}X^{2n}$
2) Let $A\in M_{n\times n}$ what is range of values of $cos(A)$
Z K Y
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loup blanc
3603 posts
loup blanc
#2 h  Apr 12, 2025, 10:46 AM
Y by
For an almost complete proof, see
https://artofproblemsolving.com/community/c15267h2880058_cosine_over_the_matrices
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