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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Aug 1, 2025
CONGRATULATIONS to all the competitors at this year’s International Mathematical Olympiad (IMO)! The US Team took second place with 5 gold medals and 1 silver - we are proud to say that each member of the 2025 IMO team has participated in an AoPS WOOT (Worldwide Online Olympiad Training) class!

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0 replies
jwelsh
Aug 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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2025China Western Mathematical Olympiad Q1
sqing   0
7 minutes ago
Source: China
Let $ABCDE$ be a convex pentagon with $\angle ACB=\angle ADE,\angle BAC+\angle CDE=180^\circ $ and $ \angle DAE+\angle BCD=180^\circ $. Prove that $AB =AE. $
0 replies
2 viewing
sqing
7 minutes ago
0 replies
J
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two interesting problem
xyz123456   1
N 11 minutes ago by xyz123456
$problem1:if\ P\left(x\right)\in \ Z\left[x\right]{,}prove\ that:\exists \ \infty \ x{,}s.t.\ \frac{x!}{P\left(x\right)}\in \ Z$
$problem 2:ifP\left(x\right)\in \ Z\left[x\right]{,}\deg \ P\equiv \ 1\left(\mod 2\right).prove\ that:\exists \ \infty \ x{,}y{,}s.t.\ \Omega \ \left(P\left(x\right)\right)\equiv \ 1\left(\mod 2\right)\Omega \ \left(P\left(y\right)\right)\equiv \ 0\left(\mod 2\right)$
1 reply
xyz123456
13 minutes ago
xyz123456
11 minutes ago
J
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KTOM Mock OSP P2
SYBARUPEMULA   2
N 31 minutes ago by SYBARUPEMULA
Source: KTOM Mock OSP 2025
Determine the largest real number $M$ such that
$$|\frac{a + b^{2025}}{b^{2025}}| + |\frac{b + c^{2025}}{c^{2025}}| + |\frac{c + d^{2025}}{d^{2025}}| + |\frac{d + e^{2025}}{e^{2025}}| + |\frac{e + a^{2025}}{a^{2025}}| > M$$holds for every non-zero integer $a, b, c, d, e.$
2 replies
1 viewing
SYBARUPEMULA
an hour ago
SYBARUPEMULA
31 minutes ago
J
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Inspired by USAMO 2003
sqing   0
32 minutes ago
Source: Own
Let $ a,b,c>0,(a+b+1)\left(\frac{1}{a} + \frac{1}{b} +1\right)= 10. $ Prove that
$$\dfrac{(2a + b + 1)^2}{2a^2 + (b + 1)^2} + \dfrac{(2b +a+ 1)^2}{2b^2 + (a + 1)^2} + \dfrac{(a+b +2)^2}{ (a + b)^2+2} \leq \frac{132 }{17}$$
0 replies
1 viewing
sqing
32 minutes ago
0 replies
J
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KTOM Mock OSP P1
SYBARUPEMULA   1
N an hour ago by Rain_L
Source: KTOM Mock OSP 2025
Given rectangle $ABCD$ and point $E$ on $AC$. The tangent line of the circumcircle $CDE$ through $D$ intersects $AB$ at point $F$. Prove that the intersection of $EF$ and $BC$ lies on the circumcircle $CDE$.
1 reply
SYBARUPEMULA
an hour ago
Rain_L
an hour ago
J
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IMO Shortlist 2014 N4
hajimbrak   76
N an hour ago by OronSH
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)

Proposed by Hong Kong
76 replies
hajimbrak
Jul 11, 2015
OronSH
an hour ago
J
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KTOM Mock OSP P3
SYBARUPEMULA   0
an hour ago
Source: KTOM Mock OSP 2025
Given odd prime $p$. For every positive integer $k \le p$, define $f(k)$ as the number of positive integers $d$ with $k \le d \le p$ and $d$ divides $kp + 1$. Determine the value of
$$f(1) + f(2) + ... + f(p).$$
0 replies
SYBARUPEMULA
an hour ago
0 replies
J
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Combinatorics
slimshady360   1
N an hour ago by Tintarn
Source: IMO 2005-Problem 6.
Does anybody have any idea how to solve this with graph theory?
1 reply
slimshady360
Aug 2, 2025
Tintarn
an hour ago
J
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The least tight inequality in South Korea
whwlqkd   1
N 2 hours ago by NTstrucker
Source: 2025 Korea Summer Camp P1(Senior)
$n\ge 3$ is an integer. $n$ real numbers $a_1,a_2,…,a_n$ satisfies $a_1+a_2+…+a_n=0$. Prove that there is 3 different integers $i,j,k$ such that $3a_i^3a_j+2a_j^3a_k+a_k^3a_i\le 0$.
1 reply
whwlqkd
2 hours ago
NTstrucker
2 hours ago
J
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a problem about factorial representation
Laan   1
N 2 hours ago by phonghatemath
Let $n$ be a positive integer greater than 1, and consider a sequence $a_1 , a_2 , \dotsc , a_k $ of positive integers such that:
$n! = \sum_{i=1}^k a_k^{n+1}$. Prove that there are less than $\frac{n+1}{2}$ distinct values in the sequence.
1 reply
Laan
2 hours ago
phonghatemath
2 hours ago
J
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120 degrees and 3…How is it related…
whwlqkd   1
N 2 hours ago by NTstrucker
Source: 2025 Korea Summer Camp P4(Senior)
$ABCD$ is a convex quadrilateral such that $\angle ABC=\angle ADC=120^{\circ}$. $M$ is the midpoint of $AC$. $X$,$Y$ are point on ray $MB,MD$ such that $MX=3MB$ and $MY=3MD$. P is the intersection of internal angle bisector of $\angle ABC$,$\angle ADC$. If $P$ lies inside of $ABCD$, prove that $\angle BPX=\angle DPY$.
1 reply
whwlqkd
2 hours ago
NTstrucker
2 hours ago
J
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Querying polynomials
ThatApollo777   1
N 2 hours ago by Tintarn
Source: India-Iran-Singapore-Taiwan 2025 P5
Alice has a secret monic degree $2025$ polynomial $P(x)$ with all coefficients in $\{0, 1\}$. Bob wants to find Alice’s polynomial but the only queries he is allowed to ask is to ask if an integer $k \in \{P(1), P(2) \dots \}$. Find the minimum number of queries he needs to ask to guarantee finding Alice’s polynomial.
1 reply
ThatApollo777
Aug 3, 2025
Tintarn
2 hours ago
J
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D1059 : A general result on prime number
Dattier   1
N 2 hours ago by Dattier
Source: les dattes à Dattier
Let $n\in \mathbb N,n>2$.
$A$ set $A$ is 'neat' (from the French partie chouette) if $A$ is a subset of $[\sqrt n, n]\cap N$ and for all pairs $(a,b)$ in $A$ with $a \neq b, \gcd(a,b)=1$.

Is it true that the maximum size that a 'neat' set can have is $\pi(n)$ ?
1 reply
Dattier
Aug 4, 2025
Dattier
2 hours ago
J
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Equal lengths and concurrency on circle
mofumofu   26
N 2 hours ago by ItsBesi
Source: Japan Mathematical Olympiad Finals 2018 Q2
Given a scalene triangle $\triangle ABC$, $D,E$ lie on segments $AB,AC$ respectively such that $CA=CD, BA=BE$. Let $\omega$ be the circumcircle of $\triangle ADE$. $P$ is the reflection of $A$ across $BC$, and $PD,PE$ meets $\omega$ again at $X,Y$ respectively. Prove that $BX$ and $CY$ intersect on $\omega$.
26 replies
mofumofu
Feb 13, 2018
ItsBesi
2 hours ago
J
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J
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An algorithm for discovering prime numbers?
Lukaluce   4
N May 30, 2025 by alexanderhamilton124
Source: 2025 Junior Macedonian Mathematical Olympiad P3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
4 replies
Lukaluce
May 18, 2025
alexanderhamilton124
May 30, 2025
J
An algorithm for discovering prime numbers?
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Reveal topic tags
number theory
prime numbers
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G H BBookmark kLocked kLocked NReply
Source: 2025 Junior Macedonian Mathematical Olympiad P3
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Lukaluce
286 posts
Lukaluce
#1 h  May 18, 2025, 3:31 PM
Y by
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
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grupyorum
1469 posts
grupyorum
#2 h  May 18, 2025, 6:31 PM
Y by
We first show that there is an $n_0$ and $\epsilon\in\{-1,1\}$ such that for every $n\ge n_0$, $p_{n+1} = 2p_n+\epsilon$.

To see this, suppose $p_1>3$. If $p_1\equiv 1\pmod{3}$ then $p_{n+1}=2p_n-1$ must hold necessarily (otherwise $3\mid 2p_n+1$ but $p_n>3$). Likewise if $p_1\equiv -1\pmod{3}$ then $p_{n+1}\equiv 2p_n+1$ must hold. If $p_1\le 3$, then $p_j>3$ for some $j>1$, so the same argument carries through. Shifting if necessary, we will analyze the sequence $p_{n+1} =2p_n-1$ and $p_{n+1}=2p_n+1$ for $p_1>3$.

Case 1. Let $p_{n+1} = 2p_n-1$ for $n\ge 1$. Set $b_n:=p_n-1$ to obtain $b_{n+1} = 2b_n$. Iterating, we find $b_n = 2^{n-1}b_1$. Consequently, $p_n = 2^{n-1}(p_1-1)+1$. Taking $n=k(p_1-1)+1$ for suitably large $k$, Fermat's theorem asserts $2^{n-1}\equiv 1\pmod{p_1}$. So, $p_1\mid p_n$ but $p_n>p_1$, hence $p_n$ cannot be a prime.

Case 2. Let $p_{n+1}=2p_n+1$ for $n\ge 1$. Set $b_n:=p_n+1$ to obtain $p_n = 2^{n-1}(p_1+1)-1$. The same choice of $n$ ensures $p_1\mid p_n$, a contradiction.

So, no such infinite sequence exists.

Remark. This is an old Bulgarian problem (between 2003-2010 I think), though I don't remember the exact year.
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Assassino9931
1554 posts
Assassino9931
#3 h  May 29, 2025, 4:17 PM
Y by
@above Hm, haven't seen this in Bulgaria, but it is popular from Baltic Way 2004.
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TopGbulliedU
24 posts
TopGbulliedU
#4 h  May 29, 2025, 5:15 PM • 1 Y
Y by alexanderhamilton124
hahaha I was in the comp,after i got out I told everyone that nobody could solve this after the results came it was only me :-D
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alexanderhamilton124
407 posts
alexanderhamilton124
#5 h  May 30, 2025, 2:47 PM
Y by
TopGbulliedU wrote:
hahaha I was in the comp,after i got out I told everyone that nobody could solve this after the results came it was only me :-D

orz gj man
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