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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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For positive integers \( a, b, c \), find all possible positive integer values o
Jackson0423   1
N 6 minutes ago by Rayanelba
For positive integers \( a, b, c \), find all possible positive integer values of
\[ \frac{a}{b} + \frac{b}{c} + \frac{c}{a}. \]
1 reply
Jackson0423
an hour ago
Rayanelba
6 minutes ago
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Dividing Pairs
Jackson0423   1
N 9 minutes ago by ND_
Source: Own
Let \( a \) and \( b \) be positive integers.
Suppose that \( a \) is a divisor of \( b^2 + 1 \) and \( b \) is a divisor of \( a^2 + 1 \).
Find all such pairs \( (a, b) \).
1 reply
Jackson0423
an hour ago
ND_
9 minutes ago
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Number Theory Chain!
JetFire008   49
N 11 minutes ago by r7di048hd3wwd3o3w58q
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
49 replies
1 viewing
JetFire008
Apr 7, 2025
r7di048hd3wwd3o3w58q
11 minutes ago
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Inspired by my own results
sqing   1
N 11 minutes ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a^2-ab+b^2=1$ . Prove that
$$ (a+b-ab)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)\leq \frac{13}{12} $$$$ (a+b+ab)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)\leq 3$$$$ (a+b)\left( \frac{a}{b+1} + \frac{b}{a+1}+ab\right)\leq 4$$
1 reply
1 viewing
sqing
12 minutes ago
sqing
11 minutes ago
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An upper bound for Iran TST 1996
Nguyenhuyen_AG   0
34 minutes ago
Let $a, \ b, \ c$ be the side lengths of a triangle. Prove that
\[\frac{ab+bc+ca}{(a+b)^2} + \frac{ab+bc+ca}{(b+c)^2} + \frac{ab+bc+ca}{(c+a)^2} \leqslant \frac{85}{36}.\]
0 replies
Nguyenhuyen_AG
34 minutes ago
0 replies
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Sharygin 2025 CR P2
Gengar_in_Galar   5
N 36 minutes ago by NicoN9
Source: Sharygin 2025
Four points on the plane are not concyclic, and any three of them are not collinear. Prove that there exists a point $Z$ such that the reflection of each of these four points about $Z$ lies on the circle passing through three remaining points.
Proposed by:A Kuznetsov
5 replies
Gengar_in_Galar
Mar 10, 2025
NicoN9
36 minutes ago
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Kinda lookimg Like AM-GM
Atillaa   1
N an hour ago by Natrium
Show that for all positive real numbers \( a, b, c \), the following inequality always holds:
\[ \frac{ab}{b+1} + \frac{bc}{c+1} + \frac{ca}{a+1} \geq \frac{3abc}{1 + abc} \]
1 reply
Atillaa
2 hours ago
Natrium
an hour ago
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best source for inequalitys
Namisgood   1
N an hour ago by Jackson0423
I need some help do I am beginner and have completed Number theory and almost all of algebra (except inequalitys) can anybody suggest a book or resource from where I can study inequalitys
1 reply
Namisgood
an hour ago
Jackson0423
an hour ago
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Find the Maximum
Jackson0423   0
an hour ago
Source: Own.
Let \( ABC \) be a triangle with \( AB \leq AC \) and \( \angle BAC = 60^\circ \).
A point \( X \) inside triangle \( ABC \) satisfies the following conditions:
\[ XA^2 + BC^2 \leq XC^2 + AB^2 \leq XB^2 + AC^2. \]Find the maximum value of \( m \) such that
\[ \frac{XA}{AC} \geq m. \]
0 replies
Jackson0423
an hour ago
0 replies
J
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Geometry with parallel lines.
falantrng   33
N an hour ago by L13832
Source: RMM 2018,D1 P1
Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .
33 replies
falantrng
Feb 24, 2018
L13832
an hour ago
J
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Something nice
KhuongTrang   25
N 2 hours ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
25 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
2 hours ago
J
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Lord of the files
AndreiVila   2
N 2 hours ago by Rohit-2006
Source: Mathematical Minds 2024 P3
On the screen of a computer there is an $2^n\times 2^n$ board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column?

Proposed by Vlad Spătaru
2 replies
AndreiVila
Sep 29, 2024
Rohit-2006
2 hours ago
J
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Inspired by old results
sqing   6
N 2 hours ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a^2+ab+b^2=2$ . Prove that
$$ (a+b-ab)\left( \frac{1}{a+1} + \frac{1}{b+1}\right)\leq 2 $$$$ (a+b-ab)\left( \frac{a}{b+1} + \frac{2b}{a+2}\right)\leq 2 $$$$ (a+b-ab)\left( \frac{a}{b+1} + \frac{2b}{a+1}\right)\leq 4$$Let $ a,b $ be reals such that $ a^2+b^2=2$ . Prove that
$$ (a+b)\left( \frac{1}{a+1} + \frac{1}{b+1}\right)= 2 $$$$ (a+b)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)=2 $$
6 replies
sqing
Today at 2:42 AM
sqing
2 hours ago
J
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An upper bound for Viet Nam TST 2005
Nguyenhuyen_AG   0
2 hours ago
Let $a, \ b, \ c$ be the side lengths of a triangle. Prove that
\[\frac{a^3}{(a+b)^3}+\frac{b^3}{(b+c)^3}+\frac{c^3}{(c+a)^3} \leqslant \frac{9}{8}.\]Viet Nam TST 2005
0 replies
Nguyenhuyen_AG
2 hours ago
0 replies
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J
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Perpendiculars and Angle Bisectors
RobRoobiks   1
N Jun 4, 2012 by Luis González
Source: Olimpiada de Mayo
Given Triangle $ABC$, $\angle B= 2 \angle C$, and $\angle A>90^\circ$. Let $M$ be midpoint of $BC$. Perpendicular of $AC$ at $C$ intersects $AB$ at $D$. Show $\angle AMB = \angle DMC$

Click to reveal hidden text
1 reply
RobRoobiks
Jun 4, 2012
Luis González
Jun 4, 2012
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Perpendiculars and Angle Bisectors
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Reveal topic tags
geometry
ratio
projective geometry
perpendicular bisector
geometry unsolved
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Source: Olimpiada de Mayo
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RobRoobiks
361 posts
RobRoobiks
#1 h  Jun 4, 2012, 6:19 PM • 2 Y
Y by Adventure10, Mango247
Given Triangle $ABC$, $\angle B= 2 \angle C$, and $\angle A>90^\circ$. Let $M$ be midpoint of $BC$. Perpendicular of $AC$ at $C$ intersects $AB$ at $D$. Show $\angle AMB = \angle DMC$

Click to reveal hidden text
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yetti
2643 posts
yetti
#2 h  Jun 4, 2012, 7:05 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Perpendicular bisector of $BC$ cuts $AB$ at $N$. $\angle BCN = \angle CBN = 2 \angle BCA$ $\Longrightarrow$ $CA$ bisects $\angle BCN$.
Together with $CD \perp CA$ $\Longrightarrow$ cross ratio $(D, N, A, B) = -1$ is harmonic.
Since $MN \perp MB$ $\Longrightarrow$ $MN$ bisects $\angle AMD$ $\Longrightarrow$ $\angle AMB = \frac{_\pi}{^2}-\angle NMA = \frac{_\pi}{^2}-\angle NMD = \angle DMC$.
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