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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
probability
ILOVEMYFAMILY   2
N 2 minutes ago by grupyorum
Let $(X_k)$ be a sequence of independent random variables with distribution $X_k \sim \text{Exp}(k)$. Define $M_n = \min_{1 \leq k \leq n} X_k$. Prove that $$\frac{n(n+1)M_n}{\ln n} \overset{P}{\rightarrow} 0$$and $$nM_n \overset{\text{a.s}}{\rightarrow} 0$$as $n \to \infty$.
2 replies
ILOVEMYFAMILY
Oct 29, 2024
grupyorum
2 minutes ago
Complex number inequalities
jhz   9
N 17 minutes ago by mihaig
Source: 2025 CTST P18
Find the smallest real number $M$ such that there exist four complex numbers $a,b,c,d$ with $|a|=|b|=|c|=|d|=1$, and for any complex number $z$, if $|z| = 1$, then\[|az^3+bz^2+cz+d|\le M.\]
9 replies
jhz
Mar 26, 2025
mihaig
17 minutes ago
complex numbers inequality
LeXuS   6
N 17 minutes ago by mihaig
Let $x$, $y$, $z$ be 3 complex numbers with $|x|=|y|=|z|=1$. Prove that $$3 \leq |x+y-z|+|z+x-y|+|y+z-x| \leq 6$$
6 replies
LeXuS
Mar 8, 2019
mihaig
17 minutes ago
Parallel lines in two-circle configuration
Tintarn   4
N 18 minutes ago by DensSv
Source: Francophone 2024, Senior P3
Let $ABC$ be an acute triangle, $\omega$ its circumcircle and $O$ its circumcenter. The altitude from $A$ intersects $\omega$ in a point $D \ne A$ and the segment $AC$ intersects the circumcircle of $OCD$ in a point $E \ne C$. Finally, let $M$ be the midpoint of $BE$. Show that $DE$ is parallel to $OM$.
4 replies
Tintarn
Apr 4, 2024
DensSv
18 minutes ago
(x^2-y^2)/ (2x^2+1) + (y^2-z^2)/(2y^2+1)+(z^2-x^2)/ (2z^2+1)<=0
parmenides51   4
N 30 minutes ago by math-olympiad-clown
Source: Greece JBMO TST 2005 p2
Prove that for each $x,y,z \in R$ it holds that
$$\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0$$
4 replies
parmenides51
Jun 16, 2019
math-olympiad-clown
30 minutes ago
IMO 2009, Problem 2
orl   150
N 31 minutes ago by Kempu33334
Source: IMO 2009, Problem 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP = OQ.$

Proposed by Sergei Berlov, Russia
150 replies
orl
Jul 15, 2009
Kempu33334
31 minutes ago
Convergence in distribution
Ernest532   4
N 31 minutes ago by grupyorum
Let $\{X_i\}$ be i.i.d with pdf $\frac{1}{\lvert x\rvert^3}\mathbb{I}_{\{\lvert x\rvert>1\}}$. Prove that $$\frac{S_n}{\sqrt{n\log(n)}}\xrightarrow[n\to\infty]{\text{d}}\mathcal{N}(0,1).$$
4 replies
Ernest532
Jun 30, 2025
grupyorum
31 minutes ago
Next term is sum of three largest proper divisors
vsamc   22
N 39 minutes ago by thdwlgh1229
Source: 2025 IMO P4
A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.

The infinite sequence $a_1, a_2, \cdots$ consists of positive integers, each of which has at least three proper divisors. For each $n\geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.

Determine all possible values of $a_1$.

Proposed by Paulius Aleknavičius, Lithuania
22 replies
vsamc
Jul 16, 2025
thdwlgh1229
39 minutes ago
Automorphic Characteristic
KHOMNYO2   1
N an hour ago by MeKnowsNothing
Given a group G where $|G| = p + 1$ for some odd prime $p$. It is known that $p \mid |Aut(G)|$. Prove that $p$ must be in the form of $4k + 3$, where $k$ is an integer. Give a group $|G|$ as an example that satisfies the property.
1 reply
KHOMNYO2
Feb 5, 2025
MeKnowsNothing
an hour ago
sequence positive
malinger   39
N an hour ago by Assassino9931
Source: ISL 2006, A2, VAIMO 2007, P4, Poland 2007
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.

Proposed by Mariusz Skalba, Poland
39 replies
malinger
Apr 22, 2007
Assassino9931
an hour ago
Italian WinterCamps test07 Problem5
mattilgale   59
N an hour ago by Assassino9931
Source: ISL 2006, A1, AIMO 2007, TST 1, P1
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i + 1} = \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.

Proposed by Harmel Nestra, Estionia
59 replies
mattilgale
Jan 29, 2007
Assassino9931
an hour ago
SL 2015 G1: Prove that IJ=AH
Problem_Penetrator   140
N an hour ago by BS2012
Source: IMO 2015 Shortlist, G1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
140 replies
Problem_Penetrator
Jul 7, 2016
BS2012
an hour ago
Inequality
SunnyEvan   0
an hour ago
Source: Own
Let $ a,b,c>0 ,$ such that: $ abc=1 .$ Prove that :
$$ 197\cdot(\sqrt{64a^2+225}+\sqrt{64b^2+225}+\sqrt{64c^2+225}) \leq 1581\cdot(a+b+c)+5304 $$
0 replies
1 viewing
SunnyEvan
an hour ago
0 replies
Putnam 2017 A2
Kent Merryfield   29
N 4 hours ago by Assassino9931
Let $Q_0(x)=1$, $Q_1(x)=x,$ and
\[Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}\]for all $n\ge 2.$ Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.
29 replies
+1 w
Kent Merryfield
Dec 3, 2017
Assassino9931
4 hours ago
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
1079 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
Z K Y
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Moubinool
5586 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
1079 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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