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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 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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Density in Factorial Congruence
steven_zhang123   1
N 23 minutes ago by zqy648
Source: 2025 Jul-谜之竞赛 Round 2 P3
Let \(p\) be a prime. For a positive integer \(m\), denote \(\nu_p(m)\) as the unique nonnegative integer \(k\) such that \(p^k \mid m\) but \(p^{k+1} \nmid m\).
Prove that for any real \(\varepsilon > 0\), there exists a positive integer \(N\) such that for all positive integers \(n \geq N\), at least \(\left( \frac{1}{p-1} - \varepsilon \right) \cdot n\) positive integers \(m\) in \(1, 2, \cdots, n\) satisfy
\[ \frac{m!}{p^{\nu_p(m!)}} \equiv 1 \pmod{p}. \]Proposed by Dong Zhenyu
1 reply
steven_zhang123
Jul 27, 2025
zqy648
23 minutes ago
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Geometry
preatsreard   5
N 37 minutes ago by Ianis
In triangle ABC , AB=17, AC=14, points D,E,F are on BC, CA,AB respectevly such that BD:DC=CE:EA=AF:FB=1:2.
If AFDE is cyclic, find the length of the side BC.
5 replies
preatsreard
Yesterday at 5:00 PM
Ianis
37 minutes ago
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Inequality
SunnyEvan   0
40 minutes ago
Source: Own
Let $ a,b,c>0 ,$ such that: $ abc=1 .$ Prove that :
$$ \sqrt{64a^2+225}+\sqrt{64b^2+225}+\sqrt{64c^2+225} \leq (7\sqrt3-4)(a+b+c)+21(3-\sqrt3) $$
0 replies
SunnyEvan
40 minutes ago
0 replies
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One of the Craziest Problem I've ever seen (see the proposers)
EthanWYX2009   3
N 41 minutes ago by zqy648
Source: 2024 September 谜之竞赛-3, by dzy&wcj&jc
For a positive integer \( n \), let \( f(n) \) be the minimal positive integer, such that for any \( n \) positive integers $x_1$, $x_2$, $\cdots$, $x_n$, \(\nu_2 \left(\sum_{i \in I} x_i\right)\) takes at most \( f(n) \) distinct integer values as \( I \) ranges over all non-empty subsets of \(\{1, 2, \cdots, n\}\).

Determine the value of \(\lim\limits_{n \to \infty} \dfrac{f(n)}{n \log_2 n}\).

Proposed by Zhenyu Dong from Hangzhou Xuejun High School, Chunji Wang from Shanghai High School, and Cheng Jiang from Tsinghua University
3 replies
EthanWYX2009
Jul 17, 2025
zqy648
41 minutes ago
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Max weight tree game
v_Enhance   6
N an hour ago by bin_sherlo
Source: USA TSTST 2025/6
Alice and Bob play a game on $n$ vertices labelled $1, 2, \dots, n$. They take turns adding edges $\{i, j\}$, with Alice going first. Neither player is allowed to make a move that creates a cycle, and the game ends after $n-1$ total turns.
Let the weight of the edge $\{i, j\}$ be $|i - j|$, and let $W$ be the total weight of all edges at the end of the game. Alice plays to maximize $W$ and Bob plays to minimize $W$. If both play optimally, what will $W$ be?

Max Lu, Kevin Wu
6 replies
v_Enhance
Jul 1, 2025
bin_sherlo
an hour ago
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IMO ShortList 1998, algebra problem 2
orl   40
N an hour ago by heheman
Source: IMO ShortList 1998, algebra problem 2
Let $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that

\[ \frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}. \]
40 replies
orl
Oct 22, 2004
heheman
an hour ago
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IMO Shortlist 2011, G4
WakeUp   135
N 2 hours ago by LHE96
Source: IMO Shortlist 2011, G4
Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear.

Proposed by Ismail Isaev and Mikhail Isaev, Russia
135 replies
WakeUp
Jul 13, 2012
LHE96
2 hours ago
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Find all natural numbers n
Iwanttostudymathbetter   2
N 2 hours ago by Feita
Find all natural numbers $n$ that satisfy:
$\sigma(n)-\varphi(n)=n+4$
2 replies
Iwanttostudymathbetter
Mar 9, 2025
Feita
2 hours ago
J
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n+1 subsets
sturdyoak2012   3
N 2 hours ago by ostriches88
Suppose you have $n+1$ subsets of $\{1, 2, \ldots, n\}$ such that any two subsets have an intersection size of exactly one. Show that two of these subsets must be the same.
3 replies
sturdyoak2012
Sep 20, 2020
ostriches88
2 hours ago
J
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Third degree and three variable system of equations
MellowMelon   59
N 2 hours ago by lpieleanu
Source: USA TST 2009 #7
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 = 3x-12y+50, \\ y^3 = 12y+3z-2, \\ z^3 = 27z + 27x. \end{cases}\]

Razvan Gelca.
59 replies
MellowMelon
Jul 18, 2009
lpieleanu
2 hours ago
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About APMO 2025
AlexanderWangUSA   4
N 2 hours ago by ehuseyinyigit
When will the APMO 2025 problems be posted?
4 replies
AlexanderWangUSA
4 hours ago
ehuseyinyigit
2 hours ago
J
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Simple geometry
LuxusN   3
N 2 hours ago by historypasser-by
Let triangle $ABC$ with orthocenter $H$ and circumcenter $O$, $AH \cap BC=D$.. Let $M$ be the midpoint $BC$, $MH$ meets $(O)$ at $N$. The line parallel to $AO$ through H meets $BC$ at $P$. Prove that $(DNP)$ is tangent to $(ABC)$.IMAGE
3 replies
LuxusN
5 hours ago
historypasser-by
2 hours ago
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functional equation
COCBSGGCTG3   15
N 2 hours ago by grupyorum
Source: Azerbaijan Senior Math Olympiad Training TST 2025 P2
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the following equality holds for any real numbers $x$ and $y$.
$f(f(x) + xf(y)) = xf(y + 1)$
15 replies
COCBSGGCTG3
Jul 21, 2025
grupyorum
2 hours ago
J
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Functional equation
Eul12   5
N 2 hours ago by jasperE3
Source: My creation
Any help for my problem
Let a be a positive integer. Find all increasing function f : IN---->IN such that f(f(n)) = (a^2)*n
for all positive integer n.
5 replies
Eul12
Jul 27, 2025
jasperE3
2 hours ago
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Loop of Logarithms
scls140511   13
N Jul 20, 2025 by Just1
Source: 2024 China Round 1 (Gao Lian)
Round 1

1 Real number $m>1$ satisfies $\log_9 \log_8 m =2024$. Find the value of $\log_3 \log_2 m$.
13 replies
scls140511
Sep 8, 2024
Just1
Jul 20, 2025
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Loop of Logarithms
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Reveal topic tags
logarithms
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Source: 2024 China Round 1 (Gao Lian)
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scls140511
115 posts
scls140511
#1 h  Sep 8, 2024, 12:48 PM • 1 Y
Y by sodiccha
Round 1

1 Real number $m>1$ satisfies $\log_9 \log_8 m =2024$. Find the value of $\log_3 \log_2 m$.
This post has been edited 4 times. Last edited by scls140511, Sep 8, 2024, 2:23 PM
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Aaronjudgeisgoat
939 posts
Aaronjudgeisgoat
#2 h  Sep 8, 2024, 1:05 PM • 4 Y
Y by MihaiT, entropybagel, aidan0626, sami1618
Click to reveal hidden text

kinda iffy but heres my sol
This post has been edited 1 time. Last edited by Aaronjudgeisgoat, Sep 8, 2024, 1:05 PM
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student1212
18 posts
student1212
#3 h  Sep 20, 2024, 5:33 AM
Y by
now from the question lets start taking log function i . e , $\log_3 \log_2 m$
after solving we would get 4049 as ans
This post has been edited 1 time. Last edited by student1212, Sep 20, 2024, 5:34 AM
Reason: .
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AshAuktober
1058 posts
AshAuktober
#4 h  Sep 20, 2024, 4:02 PM • 2 Y
Y by WuQingjia, sami1618
Note that $m = 8^{9^{2024}}$, so $$\log_2{m} = 3\cdot9^{2024} = 3^{2 \cdot 2024 + 1}$$$$\implies \log_3{\log_2{m}} = \log_3{3^{2 \cdot 2024 + 1}} = \boxed{4049}.$$
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AMM11374
2 posts
AMM11374
#5 h  Oct 16, 2024, 8:16 AM
Y by
that seems so silly
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Neeka_s08
7 posts
Neeka_s08
#8 h  Dec 21, 2024, 4:29 PM • 6 Y
Y by sami1618, HEGZAM_BANDS, Ba_2008, Zhinaaf08, Ida_m, ereh
This my solution
Attachments:
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Topiary
25 posts
Topiary
#10 h  Mar 18, 2025, 9:56 PM
Y by
The equation is equivalent to $\log_8 m = 9^{2024} \equiv \log_8 3^{4049}\rightarrow m = 8^{3^{4048}} \rightarrow m = 2^{3^4049} \rightarrow \log_3 \log_2 m = \boxed{4049}$
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ohiorizzler1434
949 posts
ohiorizzler1434
#11 h  Mar 18, 2025, 10:14 PM
Y by
Bruh! How is this even from the China TSTs? This is too trivial.
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epl1
169 posts
epl1
#12 h  Mar 18, 2025, 10:16 PM
Y by
sol
Also, what kind of competition was this problem from?
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mpcnotnpc
92 posts
mpcnotnpc
#13 h  Mar 19, 2025, 2:59 PM • 1 Y
Y by ereh
ohiorizzler1434 wrote:
Bruh! How is this even from the China TSTs? This is too trivial.

Gao lian is the very first part of the test, which means its equivalent to America's AMCs.
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SomeonecoolLovesMaths
3457 posts
SomeonecoolLovesMaths
#14 h  Mar 19, 2025, 3:12 PM
Y by
Storage
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ohiorizzler1434
949 posts
ohiorizzler1434
#15 h  Mar 19, 2025, 10:36 PM
Y by
mpcnotnpc wrote:
ohiorizzler1434 wrote:
Bruh! How is this even from the China TSTs? This is too trivial.

Gao lian is the very first part of the test, which means its equivalent to America's AMCs.

Ohhhh! Thanks for clarifying!
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yyhloveu1314
16 posts
yyhloveu1314
#16 h  Mar 23, 2025, 3:30 AM
Y by
$\log_3 \log_2 m$
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Just1
57 posts
Just1
#17 h  Jul 20, 2025, 8:47 AM
Y by
let's do it by simple math without any logarithmic formulas. $\log_8(m)=n$ and $\log_9(n)=k$ Then if we do it again in terms of $m$ and $k$ we get:$m=8^{9^k}$ and it is equal to $m=2^{3*9^k}$ so $\log_2(m)=3*9^k$ and $\log_3(3*9^k)=1+2k$ so $\log_3 \log_2 m=2k+1$ meanwhile $\log_9 \log_8 m =2024=k$ So $2k+1=4049$
Answer : $4049$
This post has been edited 3 times. Last edited by Just1, Jul 20, 2025, 8:52 AM
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