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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:
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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
Elegant but Hard Geometry
EthanWYX2009   0
26 minutes ago
Source: 2025 June 谜之竞赛-6
Two circles \( \Omega \) and \( \omega \) on the plane are internally tangent at the point \( Y \), with \( \omega \) inside \( \Omega \). Let \( A, X, C, Z, B, Y \) be six points on \( \Omega \) arranged in order, such that the lines \( AB, AC, XY, XZ \) are tangent to \( \omega \). Let \( S \) be the center of \( \omega \), and \( R \) be a point on the arc \( AX \) of \( \Omega \) not containing \( \omega \), such that \( RS \) bisects \( \angle ARX \).

Show that the lines \( BY, CZ, RT \) are concurrent.

Created by Youcheng Wang
0 replies
EthanWYX2009
26 minutes ago
0 replies
Find a<b<c<d with (a,c), (b,d), (a,d) in S
MellowMelon   17
N 36 minutes ago by ihategeo_1969
Source: USA TST 2011 P8
Let $n \geq 1$ be an integer, and let $S$ be a set of integer pairs $(a,b)$ with $1 \leq a < b \leq 2^n$. Assume $|S| > n \cdot 2^{n+1}$. Prove that there exists four integers $a < b < c < d$ such that $S$ contains all three pairs $(a,c)$, $(b,d)$ and $(a,d)$.
17 replies
MellowMelon
Jul 26, 2011
ihategeo_1969
36 minutes ago
IMO Shortlist 2010 - Problem G1
Amir Hossein   138
N an hour ago by mahyar_ais
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$

Proposed by Christopher Bradley, United Kingdom
138 replies
1 viewing
Amir Hossein
Jul 17, 2011
mahyar_ais
an hour ago
Two lengths are equal
62861   32
N an hour ago by OronSH
Source: IMO 2015 Shortlist, G5
Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.

Proposed by El Salvador
32 replies
62861
Jul 7, 2016
OronSH
an hour ago
Number Theory Marahon
Jupiterballs   15
N an hour ago by ItzsleepyXD
Let's start a number theory marathon
Rules:-
just don't post >2 problems before a solution and be friendly :)

I'll start
P1
15 replies
Jupiterballs
Jun 23, 2025
ItzsleepyXD
an hour ago
bounded or all n?
X.Allaberdiyev   2
N an hour ago by young_desi
Source: IMSC Problem 4
Determine all $n$ such that it is possible to find a convex $n$-gon which can be tiled with triangles having angles $15^\circ$, $75^\circ$ and $90^\circ$.
2 replies
X.Allaberdiyev
Jul 7, 2025
young_desi
an hour ago
Parallel lines (extension of previous problem)
RANDOM__USER   0
2 hours ago
Source: Own
Let \(D\) be an arbitrary point on the side \(BC\) in a triangle \(\triangle{ABC}\). Let \(E\) and \(F\) be the intersection of the lines parallel to \(AC\) and \(AB\) through \(D\) with \(AB\) and \(AC\). Let \(G\) be the intersection of \((AFE)\) with \((ABC)\). Let \(M\) be the midpoint of \(BC\) and \(X\) the intersection of \(AM\) with \((ABC)\). Let \(H\) be the intersection of \((XMG)\) with \(BC\). Prove that \(EF\) is parallel to \(AH\).

IMAGE

Note: This is another property of a configuration I posted before where one needed to prove that \(X, D\) and \(G\) are collinear. There are surprisingly many properties in the configuration posted earlier :P
0 replies
RANDOM__USER
2 hours ago
0 replies
CGMO8: How many k such that 2012 divides nCr(2012,k)
v_Enhance   18
N 2 hours ago by SomeonecoolLovesMaths
Source: 2012 China Girl's Mathematical Olympiad
Find the number of integers $k$ in the set $\{0, 1, 2, \dots, 2012\}$ such that $\binom{2012}{k}$ is a multiple of $2012$.
18 replies
v_Enhance
Aug 13, 2012
SomeonecoolLovesMaths
2 hours ago
AOPS MO discussions
MathMaxGreat   1
N 2 hours ago by MathMaxGreat
MathMaxGreat
3 hours ago
MathMaxGreat
2 hours ago
IMO Genre Predictions
ohiorizzler1434   112
N 2 hours ago by compoly2010
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
112 replies
ohiorizzler1434
May 3, 2025
compoly2010
2 hours ago
Matrix equation
Natrium   0
2 hours ago
If $A$ is a complex matrix with $AA^*A=A^3,$ prove that $A$ is self-adjoint, i.e., that $A^*=A.$
0 replies
Natrium
2 hours ago
0 replies
Limit of the sum of the terms of a nonlinear recurrence relation
F_Adrien   1
N Yesterday at 11:38 PM by alexheinis
Source: I am not aware of any source related to this problem. It comes from an optimization problem with bilinear constraints.
There are two questions:
[list=1]
[*] For any positive integer $n$, let $u_1 = 1$ and for any $i \in \{1, 2, \ldots, n - 1\}$ define
\[ u_{i + 1} = \frac{u_i}{1 + n u_i^2}, \]then prove that
\[ \lim_{n \to +\infty} \sum_{i = 1}^n u_i = \sqrt{3}. \]
[*] (More challenging) For any positive integer $n$, let $u_1 = 1$ and for any $i \in \{1, 2, \ldots, n - 1\}$ define
\[ u_{i + 1} = \frac{u_i}{1 + (n - i) u_i^2}, \]then prove that
\[ \lim_{n \to +\infty} \sum_{i = 1}^n u_i = 1 + \frac{\pi}{4}. \][/list]

I would be happy if anyone can provide a reference for the above (or similar) results, if any exists. Thank you in advance.
1 reply
F_Adrien
Yesterday at 6:57 PM
alexheinis
Yesterday at 11:38 PM
Brief Q about a Symmetry Argument in Triple Integral
hnkevin42   2
N Yesterday at 7:12 PM by hnkevin42
Hi, having trouble understanding how by symmetry $$\iiint_{[0,1]^3}\frac{x^2y^2z^2}{x^2y^2+y^2z^2+x^2z^2} = 3\iiint_{[0,1]^3}\frac{x^4y^2z^2}{y^2+y^2z^2+z^2}.$$Tried experimenting with a cyclic sum of the RH integrand but couldn't get anywhere due to reduction of degree with the single squared terms in denominator. Explanations much appreciated!
2 replies
hnkevin42
Yesterday at 8:14 AM
hnkevin42
Yesterday at 7:12 PM
Calculus problem
We2592   1
N Yesterday at 5:34 PM by P0tat0b0y
Q) using mean value theorem prove that $e^{\pi}>{\pi}^{e}$
1 reply
We2592
Yesterday at 4:36 PM
P0tat0b0y
Yesterday at 5:34 PM
Functions
mclolikoi   3
N Jun 2, 2025 by Mathzeus1024
Let us consider f as the following function : $ f(x)= \frac {x - \sqrt{2}}{ [x  \sqrt{2} ] } $

1- Find the definition domain $ D_f $

2-Prove that $ f $ is continous on $ \sqrt{2} $

3-Study the continuity of $ f $ on $ \frac {3 \sqrt{2} }{2} $

4-Then draw the geometric representation of $ f $ on $ ] \frac {1}{ \sqrt{2} } ; 2 \sqrt{2} [ $
3 replies
mclolikoi
Sep 23, 2012
Mathzeus1024
Jun 2, 2025
Functions
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mclolikoi
51 posts
#1 • 2 Y
Y by Adventure10, Mango247
Let us consider f as the following function : $ f(x)= \frac {x - \sqrt{2}}{ [x  \sqrt{2} ] } $

1- Find the definition domain $ D_f $

2-Prove that $ f $ is continous on $ \sqrt{2} $

3-Study the continuity of $ f $ on $ \frac {3 \sqrt{2} }{2} $

4-Then draw the geometric representation of $ f $ on $ ] \frac {1}{ \sqrt{2} } ; 2 \sqrt{2} [ $
Z K Y
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Mathzeus1024
1054 posts
#2
Y by
1) $D_{f} = x \in (-\infty, 0) \cup \left[\frac{1}{\sqrt{2}},\infty\right)$.

2) $\lim_{x \rightarrow \sqrt{2}+} f(x) = \lim_{x \rightarrow \sqrt{2}-} f(x) = 0 \Rightarrow f$ is continuous at $x=\sqrt{2}$.

3) $\lim_{x \rightarrow \frac{3\sqrt{2}}{2}+} f(x) = \frac{1}{3\sqrt{2}} \neq \lim_{x \rightarrow \frac{3\sqrt{2}}{2}-} f(x)= \frac{1}{2\sqrt{2}} \Rightarrow f$ is not continuous at $x = \frac{3\sqrt{2}}{2}$.

4)
Attachments:
This post has been edited 7 times. Last edited by Mathzeus1024, Jun 2, 2025, 1:55 PM
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Moubinool
5584 posts
#3
Y by
mclolikoi wrote:
Let us consider f as the following function : $ f(x)= \frac {x - \sqrt{2}}{ [x  \sqrt{2} ] } $

the denominator [ ...] is it floor function of $ x  \sqrt{2}  $?
Z K Y
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Mathzeus1024
1054 posts
#4
Y by
Moubinool wrote:
mclolikoi wrote:
Let us consider f as the following function : $ f(x)= \frac {x - \sqrt{2}}{ [x  \sqrt{2} ] } $

the denominator [ ...] is it floor function of $ x  \sqrt{2}  $?

My bad (I'm used to $\lfloor x \rfloor$ notation) ......corrections made above. :oops_sign:
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