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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 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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Another two parallels
jayme   2
N 4 minutes ago by jayme
Dear Mathlinkers,

1. ABCD a square
2. (A) the circle with center at A passing through B
3. P the points of intersection of the segment AC and (A)
4. I the midpoint of AB
5. Q the point of intersection of the segment IC and (A)
6. M the foot of the perpendicular to (AB) through P.
7. Y the point of intersection of the segment MC and (A)
8. X the point of intersection de AY and BC.

Prove : QX is parallel to AB.

Jean-Louis
2 replies
jayme
Today at 9:21 AM
jayme
4 minutes ago
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Diophantine equation !
ComplexPhi   9
N 21 minutes ago by MATHS_ENTUSIAST
Determine all triples $(m , n , p)$ satisfying :
\[n^{2p}=m^2+n^2+p+1\]
where $m$ and $n$ are integers and $p$ is a prime number.
9 replies
+1 w
ComplexPhi
Feb 4, 2015
MATHS_ENTUSIAST
21 minutes ago
J
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Cool inequality
giangtruong13   1
N 31 minutes ago by grupyorum
Source: Hanoi Specialized School’s Practical Math Entrance Exam (Round 2)
Let $a,b,c$ be real positive numbers such that: $a^2+b^2+c^2=4abc-1$. Prove that: $$a+b+c \geq \sqrt{abc}+2$$
1 reply
giangtruong13
2 hours ago
grupyorum
31 minutes ago
J
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Primes and sets
mathisreaI   39
N 32 minutes ago by awesomehuman
Source: IMO 2022 Problem 3
Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.
39 replies
mathisreaI
Jul 13, 2022
awesomehuman
32 minutes ago
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Interesting number theory
giangtruong13   1
N 32 minutes ago by grupyorum
Source: Hanoi Specialized School’s Practical Math Entrance Exam (Round 2)
Let $a,b$ be integer numbers $\geq 3$ satisfy that:$a^2=b^3+ab$. Prove that:
a) $a,b$ are even
b) $4b+1$ is a perfect square number
c) $a$ can’t be any power $\geq 1$ of a positive integer number
1 reply
giangtruong13
an hour ago
grupyorum
32 minutes ago
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function
CarlFriedrichGauss-1777   4
N 37 minutes ago by jasperE3
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that:
$f(2021+xf(y))=yf(x+y+2021)$
4 replies
CarlFriedrichGauss-1777
Jun 4, 2021
jasperE3
37 minutes ago
J
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Find all functions f with f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y)
Martin N.   10
N an hour ago by jasperE3
Source: (4th Middle European Mathematical Olympiad, Individual Competition, Problem 1)
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have
\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]
10 replies
Martin N.
Sep 11, 2010
jasperE3
an hour ago
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k interesting fe
skellyrah   1
N an hour ago by jasperE3
find all functions $f :\mathbb{R} \to \mathbb{R}$ such that $$xf(x+yf(xy)) + f(f(x)) = f(xf(y))^2 + (x+1)f(x)$$
1 reply
skellyrah
an hour ago
jasperE3
an hour ago
J
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International FE olympiad P3
Functional_equation   22
N an hour ago by jasperE3
Source: IFEO Day 1 P3
Find all functions $f:\mathbb R^+\rightarrow \mathbb R^+$ such that$$f(f(x)f(f(x))+y)=xf(x)+f(y)$$for all $x,y\in \mathbb R^+$

$\textit{Proposed by Functional\_equation, Mr.C and TLP.39}$
22 replies
Functional_equation
Feb 6, 2021
jasperE3
an hour ago
J
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Arbitrary point on BC and its relation with orthocenter
falantrng   19
N an hour ago by jrpartty
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
19 replies
falantrng
Yesterday at 11:47 AM
jrpartty
an hour ago
J
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Interesting inequality
sqing   3
N an hour ago by angelazhang.zlh
Source: Own
Let $ a,b,c>0 . $ Prove that
$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$Where $ k >0. $
$$\frac{a}{b}+ \frac{16b^2}{c^2} + \frac{c}{a}\geq 5$$$$\frac{a}{b}+ \frac{ b^2}{2c^2} + \frac{c}{a}\geq \frac{5}{2} $$
3 replies
sqing
4 hours ago
angelazhang.zlh
an hour ago
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hard problem
Cobedangiu   10
N an hour ago by Cobedangiu
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
10 replies
Cobedangiu
Apr 21, 2025
Cobedangiu
an hour ago
J
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inequalities
Cobedangiu   0
an hour ago
Source: UCT
Let $a,b,c>0$ and $a+b+c=3$ Prove that:
$\sum \dfrac{\sqrt{a^2+a+1}}{2a^2+10a+9}\ge \dfrac{\sqrt{3}}{7}$
0 replies
Cobedangiu
an hour ago
0 replies
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AT // BC wanted
parmenides51   103
N an hour ago by reni_wee
Source: IMO 2019 SL G1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

(Nigeria)
103 replies
parmenides51
Sep 22, 2020
reni_wee
an hour ago
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J
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locus in equilateral , about distances from it's sides
parmenides51   2
N Sep 23, 2017 by richrow12
Source: Nordic Mathematical Contest 2014 #2
Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.
2 replies
parmenides51
Sep 23, 2017
richrow12
Sep 23, 2017
J
locus in equilateral , about distances from it's sides
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Reveal topic tags
geometry
distance
Locus
Equilateral Triangle
L
G H BBookmark kLocked kLocked NReply
Source: Nordic Mathematical Contest 2014 #2
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parmenides51
30650 posts
parmenides51
#1 h  Sep 23, 2017, 12:41 PM • 1 Y
Y by Adventure10
Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.
Z K Y
The post below has been deleted. Click to close.
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richrow12
411 posts
richrow12
#2 h  Sep 23, 2017, 4:21 PM • 2 Y
Y by Adventure10, Mango247
I get answer.
Answer
Z K Y
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richrow12
411 posts
richrow12
#3 h  Sep 23, 2017, 4:31 PM • 1 Y
Y by Adventure10
Solution
This post has been edited 1 time. Last edited by richrow12, Sep 23, 2017, 4:32 PM
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