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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 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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Facts About 2025!
Existing_Human1   244
N 2 minutes ago by ZMB038
Hello AOPS,

As we enter the New Year, the most exciting part is figuring out the mathematical connections to the number we have now temporally entered

Here are some facts about 2025:
$$2025 = 45^2 = (20+25)(20+25)$$$$2025 = 1^3 + 2^3 +3^3 + 4^3 +5^3 +6^3 + 7^3 +8^3 +9^3 = (1+2+3+4+5+6+7+8+9)^2 = {10 \choose 2}^2$$
If anyone has any more facts about 2025, enlighted the world with a new appreciation for the year


(I got some of the facts from this video)
244 replies
Existing_Human1
Jan 1, 2025
ZMB038
2 minutes ago
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two solutions
τρικλινο   5
N 9 minutes ago by τρικλινο
in a book:CORE MATHS for A-LEVEL ON PAGE 41 i found the following


1st solution


$x^2-5x=0$



$ x(x-5)=0$



hence x=0 or x=5



2nd solution



$x^2-5x=0$

$x-5=0$ dividing by x



hence the solution x=0 has been lost



is the above correct?
5 replies
τρικλινο
Yesterday at 6:20 PM
τρικλινο
9 minutes ago
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sequence infinitely similar to central sequence
InterLoop   1
N 38 minutes ago by stmmniko
Source: EGMO 2025/2
An infinite increasing sequence $a_1 < a_2 < a_3 < \dots$ of positive integers is called central if for every positive integer $n$, the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1$, $b_2$, $b_3$, $\dots$ of positive integers such that for every central sequence $a_1$, $a_2$, $a_3$, $\dots$, there are infinitely many positive integers $n$ with $a_n = b_n$.
1 reply
+3 w
InterLoop
an hour ago
stmmniko
38 minutes ago
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Three concyclic quadrilaterals
Lukaluce   1
N 44 minutes ago by InterLoop
Source: EGMO 2025 P3
Let $ABC$ be an acute triangle. Points $B, D, E,$ and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic. $\newline$
The orthocentre of a triangle is the point of intersection of its altitudes.
1 reply
+2 w
Lukaluce
an hour ago
InterLoop
44 minutes ago
J
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inqualities
pennypc123456789   0
44 minutes ago
Given positive real numbers \( x \) and \( y \). Prove that:
\[ \frac{1}{x} + \frac{1}{y} + 2 \sqrt{\frac{2}{x^2 + y^2}} + 4 \geq 4 \left( \sqrt{\frac{2}{x^2 + 1}} + \sqrt{\frac{2}{y^2 + 1}} \right). \]
0 replies
pennypc123456789
44 minutes ago
0 replies
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GCD of sums of consecutive divisors
Lukaluce   1
N an hour ago by Marius_Avion_De_Vanatoare
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < ... < c_m$ be all the positive integers smaller than $N$ that are coprime to $N$. Find all $N \ge 3$ such that
\[gcd(N, c_i + c_{i + 1}) \neq 1\]for all $1 \le i \le m - 1$.
1 reply
+2 w
Lukaluce
an hour ago
Marius_Avion_De_Vanatoare
an hour ago
J
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Arithmetic means as terms of a sequence
Lukaluce   0
an hour ago
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < ...$ of positive integers is called central if for every positive integer $n$, the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$. Show that there exists an infinite sequence $b_1, b_2, b_3, ...$ of positive integers such that for every central sequence $a_1, a_2, a_3, ...$, there are infinitely many positive integers $n$ with $a_n = b_n$.
0 replies
Lukaluce
an hour ago
0 replies
J
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postaffteff
JetFire008   18
N an hour ago by Captainscrubz
Source: Internet
Let $P$ be the Fermat point of a $\triangle ABC$. Prove that the Euler line of the triangles $PAB$, $PBC$, $PCA$ are concurrent and the point of concurrence is $G$, the centroid of $\triangle ABC$.
18 replies
JetFire008
Mar 15, 2025
Captainscrubz
an hour ago
J
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Similarity
AHZOLFAGHARI   17
N an hour ago by ariopro1387
Source: Iran Second Round 2015 - Problem 3 Day 1
Consider a triangle $ABC$ . The points $D,E$ are on sides $AB,AC$ such that $BDEC$ is a cyclic quadrilateral. Let $P$ be the intersection of $BE$ and $CD$. $H$ is a point on $AC$ such that $\angle PHA = 90^{\circ}$. Let $M,N$ be the midpoints of $AP,BC$. Prove that: $ ACD \sim MNH $.
17 replies
AHZOLFAGHARI
May 7, 2015
ariopro1387
an hour ago
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A problem with non-negative a,b,c
KhuongTrang   3
N 2 hours ago by KhuongTrang
Source: own
Problem. Let $a,b,c$ be non-negative real variables with $ab+bc+ca\neq 0.$ Prove that$$\color{blue}{\sqrt{\frac{8a^{2}+\left(b-c\right)^{2}}{\left(b+c\right)^{2}}}+\sqrt{\frac{8b^{2}+\left(c-a\right)^{2}}{\left(c+a\right)^{2}}}+\sqrt{\frac{8c^{2}+\left(a-b\right)^{2}}{\left(a+b\right)^{2}}}\ge \sqrt{\frac{18(a^{2}+b^{2}+c^{2})}{ab+bc+ca}}.}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ where $t>0.$
3 replies
KhuongTrang
Mar 4, 2025
KhuongTrang
2 hours ago
J
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Number Theory Chain!
JetFire008   52
N 2 hours ago by Anto0110
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
52 replies
JetFire008
Apr 7, 2025
Anto0110
2 hours ago
J
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Convex quad
MithsApprentice   81
N 2 hours ago by LeYohan
Source: USAMO 1993
Let $\, ABCD \,$ be a convex quadrilateral such that diagonals $\, AC \,$ and $\, BD \,$ intersect at right angles, and let $\, E \,$ be their intersection. Prove that the reflections of $\, E \,$ across $\, AB, \, BC, \, CD, \, DA \,$ are concyclic.
81 replies
MithsApprentice
Oct 27, 2005
LeYohan
2 hours ago
J
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Doubt on a math problem
AVY2024   10
N Today at 6:50 AM by Yiyj1
Solve for x and y given that xy=923, x+y=84
10 replies
AVY2024
Apr 8, 2025
Yiyj1
Today at 6:50 AM
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Website to learn math
hawa   22
N Today at 6:44 AM by RedChameleon
Hi, I'm kinda curious what website do yall use to learn math, like i dont find any website thats fun to learn math
22 replies
hawa
Apr 9, 2025
RedChameleon
Today at 6:44 AM
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I cant find one problem
tanujkundu   9
N Apr 3, 2025 by vsarg
Does anybody know which problem is about when a number is a meteor and when a number is a shooting star?
9 replies
tanujkundu
Sep 17, 2024
vsarg
Apr 3, 2025
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I cant find one problem
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tanujkundu
8 posts
tanujkundu
#1 h  Sep 17, 2024, 3:00 AM
Y by
Does anybody know which problem is about when a number is a meteor and when a number is a shooting star?
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aidan0626
1827 posts
aidan0626
#2 h  Sep 17, 2024, 3:37 AM • 1 Y
Y by vsarg
Hi, this description is very vague and we probably can't help you unless you have more information on what the problem is.
What website did you find it on?
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aaaaaaaabbbb
148 posts
aaaaaaaabbbb
#3 h  Sep 17, 2024, 8:11 AM • 1 Y
Y by vsarg
Can you give more clue? Which contest is it?

The only question I found about shooting is this one AIME
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kamuii
231 posts
kamuii
#4 h  Sep 17, 2024, 8:54 PM • 1 Y
Y by vsarg
I saw one about shooting stars in intro to counting and prob
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cyberhacker
400 posts
cyberhacker
#5 h  Sep 17, 2024, 11:26 PM
Y by
this one?
Attachments:
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aritkr
1 post
aritkr
#6 h  Dec 10, 2024, 11:36 PM • 1 Y
Y by vsarg
tanujkundu wrote:
Does anybody know which problem is about when a number is a meteor and when a number is a shooting star?

check the number theory book
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GallopingUnicorn45
320 posts
GallopingUnicorn45
#7 h  Dec 12, 2024, 4:44 PM • 1 Y
Y by vsarg
It's in the C&P Intro book, that's the only one I know of.
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tanujkundu
8 posts
tanujkundu
#8 h  Apr 2, 2025, 7:30 PM • 1 Y
Y by vsarg
Thank you, everyone! I appreciate your help.
This post has been edited 1 time. Last edited by tanujkundu, Apr 2, 2025, 7:30 PM
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sadas123
1172 posts
sadas123
#9 h  Apr 2, 2025, 8:35 PM • 1 Y
Y by vsarg
tanujkundu wrote:
Thank you, everyone! I appreciate your help.

There is a problem like that in NT.
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vsarg
251 posts
vsarg
#10 h  Apr 3, 2025, 1:47 AM
Y by
I think thares a prowabiltyh quest leik thatt
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