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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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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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3-variable inequality with min(ab,bc,ca)>=1
mathwizard888   71
N 44 minutes ago by Burak0609
Source: 2016 IMO Shortlist A1
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
Proposed by Tigran Margaryan, Armenia
71 replies
mathwizard888
Jul 19, 2017
Burak0609
44 minutes ago
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IMO Shortlist 2011, Algebra 7
orl   23
N an hour ago by bin_sherlo
Source: IMO Shortlist 2011, Algebra 7
Let $a,b$ and $c$ be positive real numbers satisfying $\min(a+b,b+c,c+a) > \sqrt{2}$ and $a^2+b^2+c^2=3.$ Prove that

\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]

Proposed by Titu Andreescu, Saudi Arabia
23 replies
orl
Jul 11, 2012
bin_sherlo
an hour ago
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Find all real functions withf(x^2 + yf(z)) = xf(x) + zf(y)
Rushil   31
N an hour ago by Jakjjdm
Source: INMO 2005 Problem 6
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.
31 replies
Rushil
Aug 23, 2005
Jakjjdm
an hour ago
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Ant wanna come to A
Rohit-2006   1
N 2 hours ago by Rohit-2006
An insect starts from $A$ and in $10$ steps and has to reach $A$ again. But in between one of the s steps and can't go $A$. Find probability. For example $ABCDCDEDEA$ is valid but $ABCDEDEDEA$ is not valid.
1 reply
Rohit-2006
2 hours ago
Rohit-2006
2 hours ago
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one cyclic formed by two cyclic
CrazyInMath   31
N 2 hours ago by juckter
Source: EGMO 2025/3
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 the 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.
31 replies
CrazyInMath
Apr 13, 2025
juckter
2 hours ago
J
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Divisibility NT FE
CHESSR1DER   10
N 2 hours ago by CHESSR1DER
Source: Own
Find all functions $f$ $N \rightarrow N$ such for any $a,b$:
$(a+b)|a^{f(b)} + b^{f(a)}$.
10 replies
CHESSR1DER
Yesterday at 7:07 PM
CHESSR1DER
2 hours ago
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Prove that x1=x2=....=x2025
Rohit-2006   8
N 2 hours ago by Project_Donkey_into_M4
Source: A mock
The real numbers $x_1,x_2,\cdots,x_{2025}$ satisfy,
$$x_1+x_2=2\bar{x_1}, x_2+x_3=2\bar{x_2},\cdots, x_{2025}+x_1=2\bar{x_{2025}}$$Where {$\bar{x_1},\cdots,\bar{x_{2025}}$} is a permutation of $x_1,x_2,\cdots,x_{2025}$. Prove that $x_1=x_2=\cdots=x_{2025}$
8 replies
Rohit-2006
Apr 9, 2025
Project_Donkey_into_M4
2 hours ago
J
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IMO 2011 Problem 4
Amir Hossein   92
N 2 hours ago by LobsterJuice
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.

Proposed by Morteza Saghafian, Iran
92 replies
Amir Hossein
Jul 19, 2011
LobsterJuice
2 hours ago
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Squares in an Octagon
kred9   1
N 2 hours ago by crazydog
Source: 2025 Utah Math Olympiad #1
A regular octagon and all of its diagonals are drawn. Find, with proof, the number of squares that appear in the resulting diagram. (The side of each square must lie along one of the edges or diagonals of the octagon.)
1 reply
kred9
Apr 5, 2025
crazydog
2 hours ago
J
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quadratic with at least 1 roots
giangtruong13   1
N 3 hours ago by CM1910
Find $m$ to satisfy that the equation $x^2+mx-1=0$ has at least 1 roots $\leq -2$
1 reply
giangtruong13
4 hours ago
CM1910
3 hours ago
J
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sequence infinitely similar to central sequence
InterLoop   18
N 3 hours ago by X.Allaberdiyev
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 $a_n = b_n$.
18 replies
InterLoop
Apr 13, 2025
X.Allaberdiyev
3 hours ago
J
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all solutions of (p,n)
Sayan   11
N 3 hours ago by L13832
Source: ItaMO 2011, P5
Determine all solutions $(p,n)$ of the equation
\[n^3=p^2-p-1\]
where $p$ is a prime number and $n$ is an integer
11 replies
Sayan
Feb 14, 2012
L13832
3 hours ago
J
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Number Theory Chain!
JetFire008   55
N 3 hours ago by Lil_flip38
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
55 replies
JetFire008
Apr 7, 2025
Lil_flip38
3 hours ago
J
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Turbo's en route to visit each cell of the board
Lukaluce   15
N 3 hours ago by X.Allaberdiyev
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
15 replies
Lukaluce
Yesterday at 11:01 AM
X.Allaberdiyev
3 hours ago
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Common Insphere
bluecarneal   1
N Feb 28, 2018 by ythomashu
Source: Spring 2005 Tournament of Towns Senior A-Level #5
Prove that if a regular icosahedron and a regular dodecahedron have a common circumsphere, then they have a common insphere.

(7 points)
1 reply
bluecarneal
Mar 24, 2015
ythomashu
Feb 28, 2018
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Common Insphere
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Reveal topic tags
geometry
3D geometry
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G H BBookmark kLocked kLocked NReply
Source: Spring 2005 Tournament of Towns Senior A-Level #5
The post below has been deleted. Click to close.
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bluecarneal
9294 posts
bluecarneal
#1 h  Mar 24, 2015, 6:30 PM • 2 Y
Y by Adventure10, Mango247
Prove that if a regular icosahedron and a regular dodecahedron have a common circumsphere, then they have a common insphere.

(7 points)
Z K Y
The post below has been deleted. Click to close.
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ythomashu
6322 posts
ythomashu
#2 h  Feb 28, 2018, 4:09 AM • 2 Y
Y by Adventure10, Mango247
No one posted a solution because it is a huge bash for circumradius inradius ratios.
Anyway, note the golden rectangles inside the icosahedron and the cube inside the dodecahedron.
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