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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
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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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IMO Shortlist 2013, Geometry #2
lyukhson   82
N an hour ago by lendsarctix280
Source: IMO Shortlist 2013, Geometry #2
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
82 replies
lyukhson
Jul 9, 2014
lendsarctix280
an hour ago
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diophantine with factorials and exponents
skellyrah   20
N 2 hours ago by maromex
find all positive integers $a,b,c$ such that $$ a! + 5^b = c^3 $$
20 replies
skellyrah
May 30, 2025
maromex
2 hours ago
J
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IMO ShortList 2001, combinatorics problem 1
orl   34
N 2 hours ago by eg4334
Source: IMO ShortList 2001, combinatorics problem 1
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.
34 replies
orl
Sep 30, 2004
eg4334
2 hours ago
J
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Perpendicularity
April   35
N 2 hours ago by Schintalpati
Source: CGMO 2007 P5
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
35 replies
April
Dec 28, 2008
Schintalpati
2 hours ago
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IMO Shortlist 2011, G1
WakeUp   46
N 3 hours ago by Kempu33334
Source: IMO Shortlist 2011, G1
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.

Proposed by Härmel Nestra, Estonia
46 replies
WakeUp
Jul 13, 2012
Kempu33334
3 hours ago
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Are all solutions normal ?
loup blanc   11
N 3 hours ago by GreenKeeper
This post is linked to this one
https://artofproblemsolving.com/community/c7t290f7h3608120_matrix_equation
Let $Z=\{A\in M_n(\mathbb{C}) ; (AA^*)^2=A^4\}$.
If $A\in Z$ is a normal matrix, then $A$ is unitarily similar to $diag(H_p,S_{n-p})$,
where $H$ is hermitian and $S$ is skew-hermitian.
But are there other solutions? In other words, is $A$ necessarily normal?
I don't know the answer.
11 replies
loup blanc
Jul 17, 2025
GreenKeeper
3 hours ago
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orang NT
KevinYang2.71   23
N 3 hours ago by cubres
Source: ISL 2024 N1
Find all positive integers $n$ with the following property: for all positive divisors $d$ of $n$, we have $d+1\mid n$ or $d+1$ is prime.
23 replies
KevinYang2.71
Jul 16, 2025
cubres
3 hours ago
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I am [not] a parallelogram
peppapig_   19
N 3 hours ago by sami1618
Source: ISL 2024/G4
Let $ABCD$ be a quadrilateral with $AB$ parallel to $CD$ and $AB<CD$. Lines $AD$ and $BC$ intersect at a point $P$. Point $X$ distinct from $C$ lies on the circumcircle of triangle $ABC$ such that $PC=PX$. Point $Y$ distinct from $D$ lies on the circumcircle of triangle $ABD$ such that $PD=PY$. Lines $AX$ and $BY$ intersect at $Q$.

Prove that $PQ$ is parallel to $AB$.

Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine
19 replies
peppapig_
Jul 16, 2025
sami1618
3 hours ago
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Group Theory resources
JerryZYang   0
4 hours ago
Can someone give me some resources for group theory. ;)
0 replies
JerryZYang
4 hours ago
0 replies
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CGMO6: Airline companies and cities
v_Enhance   16
N 4 hours ago by mudkip42
Source: 2012 China Girl's Mathematical Olympiad
There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$.
16 replies
v_Enhance
Aug 13, 2012
mudkip42
4 hours ago
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USAMO 1981 #2
Mrdavid445   11
N 4 hours ago by mudkip42
Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number of communities in this county.
11 replies
Mrdavid445
Jul 26, 2011
mudkip42
4 hours ago
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IMO Shortlist 2009 - Problem A2
April   95
N 4 hours ago by lpieleanu
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
Proposed by Juhan Aru, Estonia
95 replies
April
Jul 5, 2010
lpieleanu
4 hours ago
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Axiomatic real numbers x^0
Safal   2
N 5 hours ago by Safal
Source: Discussion
Here is an interesting question for you all:

Assume $\mathbb{R}$ is an ordered field with all field axioms holding.

$\textbf{Question:}$ Suppose $x>0$ is a real number and $0\in\mathbb{R}$(Set of real numbers). Prove that if $x^0\in\mathbb{R}$. Then it (that is $x^0$) must be $1$.

Also show same thing is true if $x<0$ is a real number.

Proof

$\textbf{Question:}$ If $z\neq 0$ be any complex number and $0\in\mathbb{C}$. Show that if $z^0\in\mathbb{C}$ , then $z^0=1$.

$\textbf{Remark:}$ Order property is not true for all complex numbers.

Hint
2 replies
Safal
Yesterday at 3:20 PM
Safal
5 hours ago
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D1054 : A measure porblem
Dattier   1
N Yesterday at 5:59 PM by greenturtle3141
Source: les dattes à Dattier
$M=\bigcup\limits_{n\in\mathbb N^*} \{0,1\}^n$, for $m \in M$, $\overline m=\{mx : x\in \{0,1\}^{\mathbb N} \}$

Let $A \subset M$ with $\forall (a,b) \in A,a\neq b$, $\overline a \cap \overline b=\emptyset$.

Is it true that $\sum\limits_{a\in A} 2^{-|a|}\leq 1$ ?

PS : for $m \in M$, $|m|$ is the length of $m$, hence $|0011|=4$
1 reply
Dattier
Yesterday at 4:36 PM
greenturtle3141
Yesterday at 5:59 PM
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A Brutal Bashy Integral from Austria Integration Bee
Silver08   1
N Jun 5, 2025 by Silver08
Source: Livestream Austria Integration Bee Spring 2025
Compute:
$$\int \frac{\cos^2(x)}{\sin(x)+\sqrt{3}\cos(x)}dx$$
1 reply
Silver08
Jun 5, 2025
Silver08
Jun 5, 2025
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A Brutal Bashy Integral from Austria Integration Bee
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Reveal topic tags
calculus
integration
L
G H BBookmark kLocked kLocked NReply
Source: Livestream Austria Integration Bee Spring 2025
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Silver08
480 posts
Silver08
#1 h  Jun 5, 2025, 11:12 PM
Y by
Compute:
$$\int \frac{\cos^2(x)}{\sin(x)+\sqrt{3}\cos(x)}dx$$
Z K Y
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This post has been deleted. Click here to see post.
Silver08
480 posts
Silver08
#2 h  Jun 5, 2025, 11:30 PM
Y by
My attempt to finding the fastest solution:
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