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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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Structure of the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$ and its application t
nayr   1
N 18 minutes ago by GreenKeeper
Let $\mathbb{F}_p^{\times} = (\mathbb{Z} / p\mathbb{Z})^{\times}$ be the unit group of $\mathbb{F}_p$. It is well known that this group is cyclic. Let $g$ be a generator of this group and consider the map $\varphi : \mathbb{F}_p^{\times} \rightarrow \mathbb{F}_p^{\times}, x\mapsto x^k$ for a fixed positive integer $k$. I know that the kernel $\ker \varphi$ has oder $d:= (p-1, k)$. By the first isomorphism theorem, $\mathbb{F}_p^{\times} / \ker \varphi \cong \operatorname{im} \varphi$. Since $\mathbb{F}_p^{\times}$ is cyclic, so are its subgroups and hence $\operatorname{im} \varphi$ is cyclic of oder $\frac{p-1}{d}$. Let $H = \operatorname{im} \varphi$. Then $\mathbb{F}_p^{\times}/H$ is cyclic too and hence we have the partition:

$$\mathbb{F}_p = \{0\} \sqcup H \sqcup s^2H \sqcup \cdots \sqcup s^{d-1}H$$
for any $s\notin H$ (for example $g$).

I am trying to use this fact to solve the following question: Show that $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ have non-trivial solution for all primes $p$. Here is my attempt:

For simplicity, we rewrite the original equation for $p>3$, as $x^3+Ay^3+Bz^3\equiv 0 \pmod{p}$ (the case $p=2,3$ is easy).

If $p\equiv 2\pmod{3}$, then everything is a cube (since the cubing map $x\,mapsto x^3$ is an anutomorphism by above) and the equation is solvable.

If $p\equiv 1\pmod{3}$, let $H:=\{x^3|x\in \mathbb{F}_p^{\times}\}$ and $sH, s^2H$ be the cosets where $s \notin H$, then we have the following cases:

Case 1: $A \in H$ or $B\in H$, Without loss of generality, assume $A=4/3$ is a cube, then $4/3=a^3$ or $4=3a^3$ and we may take $(x,y,z)=(a,-1,0)$ as our solution.

Case 2: $A \in sH$ and $B\in sH$, then $A=sa^3$ and $B=sb^3$ and we may take $(x,y,z)=(0,b,-a)$ as our solution.

Case 3: $A \in s^2H$ and $B\in s^2H$, then $A=s^2a^3$ and $B=s^2b^3$ and we may take $(x,y,z)=(0,b,-a)$ as our solution.

Case 4: $A \in sH$ and $B\in s^2H$, then $A=sa^3$ and $B=s^2b^3$. This is the case I am stuck with. If we have $s^3=1$, then we may take $(x,y,z)=(ab,b,a)$ as our solution since $1+s+s^2=0$ for $s^3=1$ and $s$ is not $1$). But it is not always possible to have both $s^3=1$ and $s\notin H$. For example, I can take $s=g^{\frac{p-1}{3}}$, then $s^3=1$, but $g^{\frac{p-1}{3}}\notin H$ iff $9\nmid p-1$.

How should I resolve case 4?
1 reply
nayr
2 hours ago
GreenKeeper
18 minutes ago
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Group Theory resources
JerryZYang   3
N Today at 4:22 AM by JerryZYang
Can someone give me some resources for group theory. ;)
3 replies
JerryZYang
Yesterday at 8:38 PM
JerryZYang
Today at 4:22 AM
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Find max(a+√b+∛c) where 0< a, b, c < 1= a+b+c.
elim   7
N Today at 2:25 AM by sqing
Find $\max_{a,\,b,\,c>0\atop a+b+c=1}(a+\sqrt{b}+\sqrt[3]{c})$
7 replies
elim
Feb 7, 2020
sqing
Today at 2:25 AM
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Perpendicularity
April   35
N Yesterday at 10:14 PM 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
Yesterday at 10:14 PM
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IMO Shortlist 2011, G1
WakeUp   46
N Yesterday at 9:38 PM 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
Yesterday at 9:38 PM
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Are all solutions normal ?
loup blanc   11
N Yesterday at 9:02 PM 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
Yesterday at 9:02 PM
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Midpoints of altitudes and concurrent cevians
darij grinberg   9
N Tuesday at 2:42 PM by nsato
Source: 2nd homework problem set of the IMO training 2004, geometry, problem 1
Let $ ABC$ be a triangle. Let $ A_1$, $ B_1$, $ C_1$ be the midpoints of its sides $ BC$, $ CA$, $ AB$, and $ A_2$, $ B_2$, $ C_2$ the midpoints of the altitudes from $ A$, $ B$, $ C$. Show that the lines $ A_1A_2$, $ B_1B_2$, and $ C_1C_2$ meet at one point.
9 replies
darij grinberg
Jul 5, 2004
nsato
Tuesday at 2:42 PM
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Steiner line and isogonal lines
flower417477   1
N Jul 22, 2025 by flower417477
$\odot O$ is the circumcircle of $\triangle ABC$,$H$ is the orthocenter of $\triangle ABC$
$D$ is an arbitrary point on $\odot O$
$E$ is the reflection point of $D$ wrt $BC$,$EH$ meet $OD$ at $F$.
$K$ is the reflection point of $A$ wrt $OH$.
$P$ is a point on $\odot O$ such that $PK$ is parallel to $BC$,$Q$ is a point on $OH$ such that $PQ$ is parallel to $EH$.
$N$ is the circumcenter of $\triangle PQK$
Prove that $AF,AN$ is a pair of isogonal lines wrt $\angle BAC$
1 reply
flower417477
Jul 18, 2025
flower417477
Jul 22, 2025
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Rectangle EFGH in incircle, prove that QIM = 90
v_Enhance   70
N Jul 21, 2025 by hectorleo123
Source: Taiwan 2014 TST1, Problem 3
Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.
70 replies
v_Enhance
Jul 18, 2014
hectorleo123
Jul 21, 2025
J
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Show that AB/AC=BF/FC
syk0526   78
N Jul 21, 2025 by Kempu33334
Source: APMO 2012 #4
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.

(Here we denote $XY$ the length of the line segment $XY$.)
78 replies
syk0526
Apr 2, 2012
Kempu33334
Jul 21, 2025
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incircle with center I of triangle ABC touches the side BC
orl   41
N Jul 21, 2025 by Kempu33334
Source: Vietnam TST 2003 for the 44th IMO, problem 2
Given a triangle $ABC$. Let $O$ be the circumcenter of this triangle $ABC$. Let $H$, $K$, $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$, $B$, $C$, respectively. Denote by $A_{0}$, $B_{0}$, $C_{0}$ the midpoints of these altitudes $AH$, $BK$, $CL$, respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$, respectively. Prove that the four lines $A_{0}D$, $B_{0}E$, $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$.)
41 replies
orl
Jun 26, 2005
Kempu33334
Jul 21, 2025
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too much tangencies these days...
kamatadu   2
N Jul 19, 2025 by mudkip42
Source: Cut The Knot
Let $\Omega$ be a circle and $\gamma_1,\gamma_2$ be circles internally tangent to $\Omega$ at $P$ and $Q$. Assume that $\gamma_1$ and $\gamma_2$ are also externally tangent at point $T$. Prove that the line through $P$ perpendicular to $PT$ meets line $QT$ on $\Omega$.
2 replies
kamatadu
Jan 20, 2023
mudkip42
Jul 19, 2025
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Chords and tangent circles
math154   28
N Jul 19, 2025 by Kempu33334
Source: ELMO Shortlist 2012, G4
Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$.

Ray Li.
28 replies
math154
Jul 2, 2012
Kempu33334
Jul 19, 2025
J
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Lots of perpendiculars; compute HQ/HR
MellowMelon   56
N Jul 18, 2025 by Kempu33334
Source: USA TST 2011 P1
In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$.

Proposed by Zuming Feng
56 replies
MellowMelon
Jul 26, 2011
Kempu33334
Jul 18, 2025
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Quadruple Binomial Coefficient Sum
P162008   3
N May 30, 2025 by pineconee
Source: Self made by my Elder brother
$\sum_{p=0}^{\infty} \sum_{r=0}^{\infty} \sum_{q=1}^{\infty} \sum_{s=0}^{p+q - 1} \frac{((-1)^{p+r+s+1})(2^{p+q-1}) \binom{p + q - s - 1}{p + q - 2s - 1}}{4^s(2p^2q + 2pqr + pq + qr)(2p + 2q + 2r + 3)}.$
3 replies
P162008
May 29, 2025
pineconee
May 30, 2025
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Quadruple Binomial Coefficient Sum
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Reveal topic tags
combinatorics
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G H BBookmark kLocked kLocked NReply
Source: Self made by my Elder brother
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P162008
262 posts
P162008
#1 h  May 29, 2025, 8:04 PM • 2 Y
Y by GA34-261, aidan0626
$\sum_{p=0}^{\infty} \sum_{r=0}^{\infty} \sum_{q=1}^{\infty} \sum_{s=0}^{p+q - 1} \frac{((-1)^{p+r+s+1})(2^{p+q-1}) \binom{p + q - s - 1}{p + q - 2s - 1}}{4^s(2p^2q + 2pqr + pq + qr)(2p + 2q + 2r + 3)}.$
This post has been edited 1 time. Last edited by P162008, May 29, 2025, 8:05 PM
Reason: Typo
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vmene
316 posts
vmene
#2 h  May 29, 2025, 8:20 PM • 1 Y
Y by GA34-261
What is this monstrosity?!?!?!?!? I have NO IDEA how to solve this but /bump ig
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tliang2000
83 posts
tliang2000
#3 h  May 30, 2025, 12:29 AM
Y by
wth………………….
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pineconee
393 posts
pineconee
#4 h  May 30, 2025, 4:28 AM
Y by
I can only simplify it to
\[\sum_{p=0}^{\infty} \sum_{r=0}^{\infty} \sum_{q=1}^{\infty} (-1)^{p+r+1}\frac{p+q}{q(2p+1)(p+r)(2p+2q+2r+3)}\]Transforming this into an integral seems useful, with
\[\frac{1}{(p+r)(2p+2q+2r+3)} = \int_{0}^{1} \int_{0}^{1} u^{p+r-1}v^{2p+2q+2r+2} \, du \, dv\]but after a lot of messy calculations I got a divergent integral so something went wrong there.
Interested to see what ideas others have
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