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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
J
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Inequality with permutations
Vlados021   10
N a minute ago by HIEME_FLORESCO
Source: 2019 Belarus Team Selection Test 7.3
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that
$$ (a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n. $$
(B. Serankou, M. Karpuk)
10 replies
Vlados021
Sep 2, 2019
HIEME_FLORESCO
a minute ago
J
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An amazing inequality
Kei0923   6
N a minute ago by Kei0923
Source: Own (2021 Mock Japan MO Finals 5)
Let $n$ be a positive integer and $a_{1},a_{2},\dots, a_{n+1}$ be positive real numbers such that $$\displaystyle {\sum_{i=1}^{n+1}}\frac{1}{a_{i}+n^2}=\frac{1}{n}.$$Prove that $$\sum_{i=1}^{n+1} \frac{1}{a_{i}^{2021}+n^{2020}} \geq \frac{1}{n^{2020}}.$$
6 replies
+1 w
Kei0923
Mar 14, 2021
Kei0923
a minute ago
J
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Prime Numbers
TRcrescent27   7
N 3 minutes ago by Goutamioqmtopper
Source: 2015 Turkey JBMO TST
Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy:
$$p+q=r(p-q)^n$$
Proposed by Şahin Emrah
7 replies
TRcrescent27
Jun 22, 2016
Goutamioqmtopper
3 minutes ago
J
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Ugly Inequality
EthanWYX2009   1
N 33 minutes ago by NTstrucker
Show that
\[\sum_{k=1}^n\frac{\phi(k)}k\ln\frac{2^k}{2^k-1}\ge 1-\frac 1{2^n}.\]
1 reply
+2 w
EthanWYX2009
2 hours ago
NTstrucker
33 minutes ago
J
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Hard inequality
JK1603JK   3
N 38 minutes ago by GeoMorocco
Source: unknown?
Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $$P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$
3 replies
JK1603JK
Yesterday at 4:24 AM
GeoMorocco
38 minutes ago
J
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Geometry with orthocenter config
thdnder   3
N an hour ago by moony_
Source: Own
Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$. Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$. Prove that the lines $DG$, $EF$, and $A$-median of $\triangle ABC$ are concurrent.
3 replies
+1 w
thdnder
Yesterday at 5:26 PM
moony_
an hour ago
J
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7^a - 3^b divides a^4 + b^2 (from IMO Shortlist 2007)
Dida Drogbier   38
N an hour ago by sansgankrsngupta
Source: ISL 2007, N1, VAIMO 2008, P4
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a - 3^b$ divides $ a^4 + b^2$.

Author: Stephan Wagner, Austria
38 replies
1 viewing
Dida Drogbier
Apr 21, 2008
sansgankrsngupta
an hour ago
J
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Choose P on (AOB) and Q on (AOC)
MarkBcc168   8
N an hour ago by NO_SQUARES
Source: ELMO Shortlist 2024 G5
Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$, respectively, such that $A$, $P$, and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$, then $\angle BTD=\angle CAP$.

Tiger Zhang
8 replies
MarkBcc168
Jun 22, 2024
NO_SQUARES
an hour ago
J
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easy functional
B1t   13
N an hour ago by AshAuktober
Source: Mongolian TST 2025 P1.
Denote the set of real numbers by $\mathbb{R}$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y, z \in \mathbb{R}$,
\[ f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)). \]
13 replies
B1t
Apr 26, 2025
AshAuktober
an hour ago
J
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problem interesting
Cobedangiu   3
N an hour ago by tom-nowy
Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
3 replies
Cobedangiu
Today at 5:06 AM
tom-nowy
an hour ago
J
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Do not try to bash on beautiful geometry
ItzsleepyXD   2
N an hour ago by moony_
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
2 replies
ItzsleepyXD
4 hours ago
moony_
an hour ago
J
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Trivial fun Equilateral
ItzsleepyXD   3
N 2 hours ago by Tsikaloudakis
Source: Own , Mock Thailand Mathematic Olympiad P1
Let $ABC$ be a scalene triangle with point $P$ and $Q$ on the plane such that $\triangle BPC , \triangle CQB$ is an equilateral . Let $AB$ intersect $CP$ and $CQ$ at $X$ and $Z$ respectively and $AC$ intersect $BP$ and $BQ$ at $Y$ and $W$ respectively .
Prove that $XY\parallel ZW$
3 replies
ItzsleepyXD
5 hours ago
Tsikaloudakis
2 hours ago
J
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Question on Balkan SL
Fmimch   3
N 2 hours ago by GreekIdiot
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
3 replies
Fmimch
Today at 12:13 AM
GreekIdiot
2 hours ago
J
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Something about (BIC)
flower417477   2
N 2 hours ago by sami1618
Given $\triangle ABC$ with incenter $I$,$D$ is a point on $BC$ ,the bisector of $\angle ADB$ meet $(BIC)$ at $E,F$.Prove that $\angle EAD=\angle IAF$
2 replies
flower417477
Monday at 3:58 PM
sami1618
2 hours ago
J
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J
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D1018 : Can you do that ?
Dattier   1
N Apr 5, 2025 by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
Apr 5, 2025
J
D1018 : Can you do that ?
G H J
Reveal topic tags
arithmetic
number theory
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G H BBookmark kLocked kLocked NReply
Source: les dattes à Dattier
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Dattier
1491 posts
Dattier
#1 h  Mar 24, 2025, 6:01 AM • 1 Y
Y by PikaPika999
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Dattier
1491 posts
Dattier
#9 h  Apr 5, 2025, 8:40 PM • 1 Y
Y by PikaPika999
No one ?
Z K Y
N Quick Reply
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