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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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Putnam 1992 B1
sqrtX   2
N 2 minutes ago by de-Kirschbaum
Source: Putnam 1992
Let $S$ be a set of $n$ distinct real numbers. Let $A_{S}$ be the set of numbers that occur as averages of two distinct
elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_{S}$?
2 replies
sqrtX
Jul 18, 2022
de-Kirschbaum
2 minutes ago
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IMO ShortList 2003, combinatorics problem 4
darij grinberg   39
N 2 hours ago by ThatApollo777
Source: Problem 5 of the German pre-TST 2004, written in December 03
Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.
39 replies
darij grinberg
May 17, 2004
ThatApollo777
2 hours ago
J
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greatest volume
hzbrl   4
N 2 hours ago by hzbrl
Source: purple comet
A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume $n \pi$. Find $n$.
4 replies
hzbrl
May 8, 2025
hzbrl
2 hours ago
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Projective geo
drmzjoseph   1
N 2 hours ago by Luis González
Any pure projective solution? I mean no metrics, Menelaus, Ceva, bary, etc
Only pappus, desargues, dit, etc
Btw prove that $X',P,K$ are collinear, and $P,Q$ are arbitrary points
1 reply
drmzjoseph
Mar 6, 2025
Luis González
2 hours ago
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2019 Iberoamerican Mathematical Olympiad, P1
jbaca   9
N 2 hours 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
2 hours ago
J
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Conditional geo with centroid
a_507_bc   7
N 2 hours 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
2 hours ago
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People live in Kansas?
jj_ca888   13
N 3 hours 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
3 hours ago
J
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Symmetric integer FE
a_507_bc   5
N 3 hours 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
3 hours ago
J
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Channel name changed
Plane_geometry_youtuber   6
N 3 hours 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
3 hours ago
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How many cases did you check?
avisioner   18
N 3 hours 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
3 hours ago
J
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Beautiful geo but i cant solve this
phonghatemath   3
N 3 hours 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
3 hours ago
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ISI UGB 2025
Entrepreneur   3
N 3 hours 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
3 hours ago
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Expand into a Fourier series
Tip_pay   1
N 5 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
5 hours ago
J
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functional analysis
ILOVEMYFAMILY   0
5 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
5 hours ago
0 replies
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J
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A very simple question about calculus for middle school students
Craftybutterfly   19
N Apr 15, 2025 by Craftybutterfly
Source: own
$\lim_{x \to 8} \frac{2x^2+13x+6}{x^2+14x+48}=$ ? (there is an easy workaround)
(I know this is very easy- a little child can solve this in 1 second kinda problem so don't argue or mock me please)
19 replies
Craftybutterfly
Apr 9, 2025
Craftybutterfly
Apr 15, 2025
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A very simple question about calculus for middle school students
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Craftybutterfly
600 posts
Craftybutterfly
#1 h  Apr 9, 2025, 7:41 AM
Y by
$\lim_{x \to 8} \frac{2x^2+13x+6}{x^2+14x+48}=$ ? (there is an easy workaround)
(I know this is very easy- a little child can solve this in 1 second kinda problem so don't argue or mock me please)
This post has been edited 2 times. Last edited by Craftybutterfly, Apr 10, 2025, 9:26 PM
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Craftybutterfly
600 posts
Craftybutterfly
#2 h  Apr 9, 2025, 7:44 AM
Y by
sol
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Yiyj1
1271 posts
Yiyj1
#3 h  Apr 9, 2025, 7:49 AM
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GentlePanda24
750 posts
GentlePanda24
#4 h  Apr 9, 2025, 2:33 PM
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solution
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KAME06
161 posts
KAME06
#5 h  Apr 9, 2025, 4:10 PM
Y by
We can plug because the function is continuous in $8$.
Pretty well known that polynomials are continuous, and $2x^2+13x+6$ and $x^2+14x+48$ are continuous on $8$ and different from $0$ when you evaluate them on $8$.
That implies that $\frac{2x^2+13x+6}{x^2+14x+48}$ is continuous on $8$, so $\lim_{x \to 8} \frac{2x^2+13x+6}{x^2+14x+48}=\frac{2(8)^2+13(8)+6}{(8)^2+14(8)+48}$ (factorize if u want to do less work).
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HacheB2031
408 posts
HacheB2031
#6 h  Apr 10, 2025, 5:22 AM
Y by
Craftybutterfly wrote:
Find $\lim_{x \to -6} \frac{2x^2+13x+6}{x^2+14x+48}.$

FTFY
This post has been edited 1 time. Last edited by HacheB2031, Apr 10, 2025, 2:37 PM
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Craftybutterfly
600 posts
Craftybutterfly
#7 h  Apr 10, 2025, 5:24 AM
Y by
HacheB2031 wrote:
Craftybutterfly wrote:
Find $\lim_{x \to -8} \frac{2x^2+13x+6}{x^2+14x+48}.$

FTFY
?
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Craftybutterfly
600 posts
Craftybutterfly
#9 h  Apr 10, 2025, 6:02 AM
Y by
@sp0rtman00000Click to reveal hidden text
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HacheB2031
408 posts
HacheB2031
#10 h  Apr 10, 2025, 2:37 PM • 1 Y
Y by LawofCosine
Craftybutterfly wrote:
HacheB2031 wrote:
Craftybutterfly wrote:
Find $\lim_{x \to -6} \frac{2x^2+13x+6}{x^2+14x+48}.$

FTFY
?
I'm sorry, I realized it should probably be as $x\to-6.$ Let me explain why.
When you make limits, you often want them to tend to a value where a function is not defined. Take the limit \[\lim_{x\to0}\frac{\sin x}x.\]Clearly, direct evaluation gives \[\frac{\sin0}0=\frac00,\]an indeterminate form, but (using methods such as the squeeze theorem) you can show that \[\lim_{x\to0}\frac{\sin x}x=1.\]Despite the function not being defined there, it has a well-defined limit. In my [edited] post, notice that as $x\to-6,$ you also have $x^2+14x+48\to0$ and $2x^2+13x+6\to0,$ making this limit not evaluatable with direct evaluation. (The limit may exist, but in the old post where $x\to-8,$ it blows up.) If you have the limit of a rational function, a function where the numerator and denominator are both polynomials, you can cancel common factors to redefine the function where it wasn't defined. Also, note that you can't always use direct evaluation. Let \[f(x)=\begin{cases}x^2&x\ne0\\1&x=0\end{cases}.\]Try to evaluate the limit \[\lim_{x\to0}f(x).\]If we directly evaluate it, we get that \[\lim_{x\to0}f(x)=f(0)=1.\]However, $f(x)$ has a problem: it isn't continuous, or, if you drew its graph, you would need to pick up your pencil from the paper. The actual limit turns out to be \[\lim_{x\to0}f(x)=\lim_{x\to0}x^2=0.\]Because $x^2$ is a polynomial, it is continuous; all polynomials are continuous. As well as all rational functions, over their domain. Anywhere a rational function is defined, it is continuous, except at points where its denominator is $0.$ sidenote
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Craftybutterfly
600 posts
Craftybutterfly
#11 h  Apr 10, 2025, 6:19 PM
Y by
It is $\lim_{x\to8}$ not $\lim_{x\to-6}$
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yaxuan
3429 posts
yaxuan
#12 h  Apr 10, 2025, 8:51 PM
Y by
op wrote:
a kindergartener can solve this in 1 second
Bruh
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Craftybutterfly
600 posts
Craftybutterfly
#13 h  Apr 10, 2025, 9:27 PM
Y by
Craftybutterfly wrote:
sol
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Oshawoot
148 posts
Oshawoot
#14 h  Apr 14, 2025, 2:27 AM
Y by
whats lim again？。。。
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LawofCosine
864 posts
LawofCosine
#15 h  Apr 14, 2025, 2:40 AM
Y by
a limit (concept in calculus)
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GentlePanda24
750 posts
GentlePanda24
#16 h  Apr 14, 2025, 1:11 PM
Y by
Craftybutterfly wrote:
a kindergartener can solve this in 1 second

I think that is a bit exagerated
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GentlePanda24
750 posts
GentlePanda24
#17 h  Apr 14, 2025, 1:12 PM
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Craftybutterfly wrote:
a kindergartener can solve this in 1 second

I think that is a bit exagerated
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leahlyoung106
63 posts
leahlyoung106
#18 h  Apr 14, 2025, 2:12 PM
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Craftybutterfly
600 posts
Craftybutterfly
#19 h  Apr 15, 2025, 5:24 AM
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Question: what is an example of a with no limit or a limit of 0
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aidan0626
1966 posts
aidan0626
#20 h  Apr 15, 2025, 5:29 AM • 1 Y
Y by LawofCosine
Craftybutterfly wrote:
Question: what is an example of a with no limit or a limit of 0

x-> -8 has no limit
x-> -1/2 gives a limit of 0
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Craftybutterfly
600 posts
Craftybutterfly
#21 h  Apr 15, 2025, 5:35 AM
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Thx @bove
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