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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
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

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0 replies
jwelsh
Jul 1, 2025
0 replies
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
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
Geometry
preatsreard   0
3 minutes ago
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.
0 replies
preatsreard
3 minutes ago
0 replies
diophantine equations
andralph   16
N 8 minutes ago by Natrium
Find all natural numbers x such that 7^x + 2 is a perfect square
16 replies
+1 w
andralph
Jul 27, 2025
Natrium
8 minutes ago
PL^2 = DQ x PQ wanted, DQ = DK, <ADB = 45^o, <KDP = 30^o given
parmenides51   4
N 9 minutes ago by WobbyBoii
Source: 2009 Balkan Shortlist BMO G1 - easy
In the triangle $ABC, \angle  BAC$ is acute, the angle bisector of $\angle  BAC$ meets $BC$ at $D, K$ is the foot of the perpendicular from $B$ to $AC$, and $\angle ADB = 45^o$. Point $P$ lies between $K$ and $C$ such that $\angle KDP = 30^o$. Point $Q$ lies on the ray $DP$ such that $DQ = DK$. The perpendicular at $P$ to $AC$ meets $KD$ at $L$. Prove that $PL^2 = DQ \cdot PQ$.
4 replies
parmenides51
Apr 5, 2020
WobbyBoii
9 minutes ago
Shortlist 2017/G5
fastlikearabbit   70
N 32 minutes ago by Aiden-1089
Source: IMO Shortlist 2017, Korea TST 2018
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
70 replies
fastlikearabbit
Jul 10, 2018
Aiden-1089
32 minutes ago
beautiful geometry
pnf   0
32 minutes ago
let $\triangle$ $ABC$ be a triangle with circumcenter $O$.$\ell$ is a fix line passing through $O$ and $p$ an arbitrary point on it . denote $\omega$ the circumcircle of pedal triangle of $p$. prove that $\omega$ passese through a fix point
0 replies
pnf
32 minutes ago
0 replies
Asian Pacific Mathematical Olympiad 2010 Problem 4
Goutham   69
N 37 minutes ago by Shan3t
Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.
69 replies
Goutham
May 7, 2010
Shan3t
37 minutes ago
Ali BAABBA
drago.7437   0
an hour ago
Source: Dimitry Fomim
Ali-Baba wants to get into Sesame cave. There is a barrel in front of the entrance. It has four holes with a jar inside each of them, and there is a herring in every jar. A herring can sit in a jar with its head up or down. Ali-Baba can stick his hands into any two holes, and, after examining the positions of the herrings, change their positions in an arbitrary manner. After this operation the barrel begins to rotate, and after it stops Ali-Baba cannot tell one hole from another. The Sesame cave will open if and only if all four herrings are in the same position. How must Ali-Baba act to get into the cave?
0 replies
drago.7437
an hour ago
0 replies
Number Theory
AnhQuang_67   1
N an hour ago by shanelin-sigma
Prove that exists a positive integer number $n$ has exactly 2000 prime divisors satisfying $n\mid 2^n+1$
1 reply
AnhQuang_67
an hour ago
shanelin-sigma
an hour ago
2024 International Math Olympiad Number Theory Shortlist, Problem 3
brainfertilzer   17
N an hour ago by SimplisticFormulas
Source: 2024 ISL N3
Determine all sequences $a_1, a_2, \dots$ of positive integers such that for any pair of positive integers $m\leqslant n$, the arithmetic and geometric means
\[ \frac{a_m + a_{m+1} + \cdots + a_n}{n-m+1}\quad\text{and}\quad (a_ma_{m+1}\cdots a_n)^{\frac{1}{n-m+1}}\]are both integers.
17 replies
brainfertilzer
Jul 16, 2025
SimplisticFormulas
an hour ago
Fresh Geometry
JARP091   1
N an hour ago by JARP091
Source: Own
A graph is defined as a set of points (\( \geq 3 \)) no three of which are collinear and lines (\( \geq 1 \)) none of which pass through a point. You are allowed to do the following operations on the graph:
(i) Reflect a point about a line (unless the point is already on the graph),
(ii) Reflect a line about a point (unless the line is already on the graph),
(iii) Join three collinear points if any.

Question:
(i) Does the game necessarily end?
(ii) Prove that it is always possible to find a set of concyclic points after a finite number of moves.
I have proved for 3 points and have checked in Geogebra for more points and it seems this is true.
1 reply
JARP091
Yesterday at 11:52 AM
JARP091
an hour ago
IMO Shortlist 2014 N1
hajimbrak   48
N an hour ago by Jupiterballs
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n  - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .

Proposed by Serbia
48 replies
hajimbrak
Jul 11, 2015
Jupiterballs
an hour ago
Strange Conditional Sequence
MarkBcc168   22
N an hour ago by Jupiterballs
Source: APMO 2019 P2
Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\geq 1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\geq 1$ we have
$$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$For each $m$, determine all possible values of $a_1$ such that every term in the sequence is an integer.
22 replies
MarkBcc168
Jun 11, 2019
Jupiterballs
an hour ago
One of the Craziest Problem I've ever seen (see the proposers)
EthanWYX2009   2
N an hour ago by R9182
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
2 replies
EthanWYX2009
Jul 17, 2025
R9182
an hour ago
Nice sequence bound
VicKmath7   5
N an hour ago by Assassino9931
Source: Bulgaria MO Regional round 2024, 12.2
Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$for all positive integers $n>N$. Show that there exists a constant $c>0$, such that $x_n \leq \frac{n} {2}+c$ for all positive integers $n$.
5 replies
VicKmath7
Feb 13, 2024
Assassino9931
an hour ago
Combi Algorithm/PHP/..
CatalanThinker   1
N May 28, 2025 by CatalanThinker
Source: Olympiad_Combinatorics_by_Pranav_A_Sriram
5. [Czech and Slovak Republics 1997]
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and
switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.
1 reply
CatalanThinker
May 28, 2025
CatalanThinker
May 28, 2025
Combi Algorithm/PHP/..
G H J
Source: Olympiad_Combinatorics_by_Pranav_A_Sriram
The post below has been deleted. Click to close.
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CatalanThinker
13 posts
#1
Y by
5. [Czech and Slovak Republics 1997]
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and
switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
CatalanThinker
13 posts
#2
Y by
Any ideas?
Z K Y
N Quick Reply
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