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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 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
J
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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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A sharp one with 3 var (3)
mihaig   5
N 23 minutes ago by JARP091
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a^2+b^2+c^2+5abc\geq8.$$
5 replies
mihaig
Tuesday at 5:17 PM
JARP091
23 minutes ago
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IMO 2016 Problem 1
quangminhltv99   152
N 36 minutes ago by MTA_2024
Source: IMO 2016
Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the midpoint of $CF$. Let $X$ be the point such that $AMXE$ is a parallelogram. Prove that $BD,FX$ and $ME$ are concurrent.
152 replies
1 viewing
quangminhltv99
Jul 11, 2016
MTA_2024
36 minutes ago
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Circles with Altitude Feet
tastymath75025   32
N an hour ago by Aiden-1089
Source: 2019 ELMO Shortlist G4
Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$.

Proposed by Daniel Hu
32 replies
tastymath75025
Jun 27, 2019
Aiden-1089
an hour ago
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Not so beautiful
m4thbl3nd3r   2
N an hour ago by m4thbl3nd3r
Let $a, b,c>0$ such that $b+c>a$. Prove that $$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$
2 replies
m4thbl3nd3r
4 hours ago
m4thbl3nd3r
an hour ago
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Concurrent lines, angle bisectors
legogubbe   1
N an hour ago by Captainscrubz
Source: ???
Hi AoPS!

Let $ABC$ be an isosceles triangle with $AB=AC$, and $M$ an arbitrary point on side $BC$. The internal angle bisector of $\angle MAB$ meets the circumcircle of $\triangle ABC$ again at $P \neq A$, and the internal angle bisector of $\angle CAM$ meets it again at $Q \neq A$. Show that lines $AM$, $BQ$ and $CP$ are concurrent.
1 reply
legogubbe
Yesterday at 11:09 PM
Captainscrubz
an hour ago
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Another NT FE
nukelauncher   63
N 2 hours ago by sansgankrsngupta
Source: ISL 2019 N4
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
63 replies
nukelauncher
Sep 22, 2020
sansgankrsngupta
2 hours ago
J
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strange geometry problem
Zavyk09   1
N 2 hours ago by Captainscrubz
Source: own
Let $ABC$ be a triangle with circumcenter $O$ and internal bisector $AD$. Let $AD$ cuts $(O)$ again at $M$ and $MO$ cuts $(O)$ again at $N$. Point $L$ lie on $AD$ such that $(AD, LM) = -1$. The line pass through $L$ and perpendicular to $AD$ intersects $NC, NB$ at $P, Q$ respectively. Let circumcircle of $\triangle NPQ$ cuts $(O)$ at $G \ne N$. Prove that $\angle AGD = 90^{\circ}$.
1 reply
Zavyk09
Yesterday at 4:32 PM
Captainscrubz
2 hours ago
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Recurrence trouble
SomeonecoolLovesMaths   1
N 2 hours ago by alexheinis
Let $0 < x_0 < y_0$ be real numbers. Define $x_{n+1} = \frac{x_n + y_n}{2}$ and $y_{n+1} = \sqrt{x_{n+1}y_n}$.
Prove that $\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n$ and hence find the limit.
1 reply
SomeonecoolLovesMaths
Yesterday at 11:27 AM
alexheinis
2 hours ago
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n^6 + 5n^3 + 4n + 116 is the product of two or more consecutive numbers
Amir Hossein   2
N 2 hours ago by KTYC
Source: Bulgaria JBMO TST 2018, Day 1, Problem 3
Find all positive integers $n$ such that the number
$$n^6 + 5n^3 + 4n + 116$$is the product of two or more consecutive numbers.
2 replies
Amir Hossein
Jun 25, 2018
KTYC
2 hours ago
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IMO Shortlist 2009 - Problem G3
April   49
N 3 hours ago by Ilikeminecraft
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.
Prove that $GR=GS$.

Proposed by Hossein Karke Abadi, Iran
49 replies
April
Jul 5, 2010
Ilikeminecraft
3 hours ago
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Four tangent lines concur on the circumcircle
v_Enhance   36
N 3 hours ago by Ilikeminecraft
Source: USA TSTST 2018 Problem 3
Let $ABC$ be an acute triangle with incenter $I$, circumcenter $O$, and circumcircle $\Gamma$. Let $M$ be the midpoint of $\overline{AB}$. Ray $AI$ meets $\overline{BC}$ at $D$. Denote by $\omega$ and $\gamma$ the circumcircles of $\triangle BIC$ and $\triangle BAD$, respectively. Line $MO$ meets $\omega$ at $X$ and $Y$, while line $CO$ meets $\omega$ at $C$ and $Q$. Assume that $Q$ lies inside $\triangle ABC$ and $\angle AQM = \angle ACB$.

Consider the tangents to $\omega$ at $X$ and $Y$ and the tangents to $\gamma$ at $A$ and $D$. Given that $\angle BAC \neq 60^{\circ}$, prove that these four lines are concurrent on $\Gamma$.

Evan Chen and Yannick Yao
36 replies
v_Enhance
Jun 26, 2018
Ilikeminecraft
3 hours ago
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3rd AKhIMO for university students, P3
UzbekMathematician   1
N 4 hours ago by pineconee
Source: AKhIMO 2025, P3
Two points are chosen randomly - independently with uniform probability - from a semicircular arc with radius 1. A third point is chosen randomly - independently with uniform probability - from the diameter that connects the endpoints of the arc. What is expected value of the area of the triangle with the three chosen points as its vertices?
1 reply
UzbekMathematician
Yesterday at 1:57 PM
pineconee
4 hours ago
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D1038 : A generalization of Jensen
Dattier   4
N Yesterday at 11:21 PM by Dattier
Source: les dattes à Dattier
Let $f \in C^1([0,1]), g \in C^2(f([0;1]))$.

Is it true that

$$\min(|g''|)\times \min(|f'|^2) \leq 24 \times\left|\int_0^1g(f(x)) \text{d}x- g(\int_0^1 f(x) \text{d}x) \right| \leq \max(|g''|)\times \max(|f'|^2)$$?
4 replies
Dattier
Yesterday at 12:15 PM
Dattier
Yesterday at 11:21 PM
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Prove the statement
Butterfly   13
N Yesterday at 10:11 PM by oty
Given an infinite sequence $\{x_n\} \subseteq [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
13 replies
Butterfly
May 7, 2025
oty
Yesterday at 10:11 PM
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How to solve this problem
xiangovo   1
N Apr 21, 2025 by loup blanc
Source: website
How many nonzero points are there on x^3y + y^3z + z^3x = 0 over the finite field \mathbb{F}_{5^{18}} up to scaling?
1 reply
xiangovo
Mar 19, 2025
loup blanc
Apr 21, 2025
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How to solve this problem
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Reveal topic tags
algebraic geometry
superior algebra
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Source: website
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xiangovo
1 post
xiangovo
#1 h  Mar 19, 2025, 11:54 AM
Y by
How many nonzero points are there on x^3y + y^3z + z^3x = 0 over the finite field \mathbb{F}_{5^{18}} up to scaling?
Z K Y
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loup blanc
3601 posts
loup blanc
#2 h  Apr 21, 2025, 11:09 AM
Y by
With a computer, I find
for $r\leq 5$, the number of such points (up to scaling) over $\mathbb{F}_{5^r}$ is $5^r+1$.
Then I conjectured that the required result is $5^{18}+1$, but this is probably false. -see below-
EDIT 1. The considered projective curve is the Klein quartic over $\mathbb{F}_q$.
Indeed, the number of its points is $q+1$ except when $q=1 \mod 7$.
Unfortunately $5^r=1 \mod 7$ when $r=6,12,18,\cdots$.
EDIT 2. Over $\mathbb{F}_{5^6}$, the number of points is $16376$ and not $5^6+1=15626$.

I think that @ xiangovo is a joker.
This post has been edited 3 times. Last edited by loup blanc, Apr 21, 2025, 5:55 PM
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