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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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Six variables inequality
JK1603JK   0
an hour ago
Source: unknown
Let $a,b,c,x,y,z>0$ then prove $$\frac{y+z}{x}\cdot \frac{a}{b+c}+\frac{z+x}{y}\cdot \frac{b}{c+a}+\frac{x+y}{z}\cdot \frac{c}{a+b}\ge 2\sqrt{1+\frac{10abc}{(a+b)(b+c)(c+a)}}.$$
0 replies
1 viewing
JK1603JK
an hour ago
0 replies
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Inspired by old results
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a^2+ab+b^2 =3 $ . Prove that
$$ \frac{53-\sqrt 3}{46} \geq\frac{1}{ 2a^2+a+1 }+ \frac{1}{2b^2+b+1 } \geq \frac{1}{2 }$$$$ \frac{3}{5 }> \frac{1}{ 2a^2+b+1 }+ \frac{1}{2b^2+a+1 } \geq \frac{1}{2 }$$$$ \frac{3}{2} \geq\frac{1}{ 2a^2+a+1 }+ \frac{1}{2b^2+b+1 } +ab\geq \frac{53-\sqrt 3}{46} $$$$ \frac{3}{2} \geq\frac{1}{ 2a^2+b+1 }+ \frac{1}{2b^2+a+1 } +ab\geq \frac{7\sqrt 3-5}{14} $$$$ \frac{53-\sqrt 3}{46} \geq \frac{1}{ 2a^2+a+1 }+ \frac{1}{2b^2+b+1 }-ab\geq -\frac{1}{2 }$$$$ \frac{7\sqrt 3-5}{14}\geq \frac{1}{ 2a^2+b+1 }+ \frac{1}{2b^2+a+1 }-ab\geq -\frac{1}{2 }$$
1 reply
sqing
an hour ago
sqing
an hour ago
J
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Interesting geometry
AlexCenteno2007   1
N 2 hours ago by MathLuis
Let the circle (I) and G be the inscribed circle (with center I) and the centroid of triangle ABC, respectively.
X and X' are the points of tangency of the inscribed circle A-mixtiline and the eccentric circle A-mixtiline with side BC of triangle ABC, respectively.
M is the midpoint of side BC.
K is the intersection of the incircle of circle (I) with BC.
AN is the height of triangle ABC from vertex A.
K' is the reflection of K with respect to M.
L is the second point of intersection between circle ABC (circle of the triangle) and the reflection of line AM with respect to line AI.

NG, XK, LM, and X'K' meet on the circle of ABC.
1 reply
AlexCenteno2007
2 hours ago
MathLuis
2 hours ago
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IMO Shortlist 2014 C4
hajimbrak   25
N 2 hours ago by math-olympiad-clown
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.

Proposed by Tamas Fleiner and Peter Pal Pach, Hungary
25 replies
hajimbrak
Jul 11, 2015
math-olympiad-clown
2 hours ago
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No more topics!
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Technical Analysis of Three Variable Inequalities
Amir Hossein   3
N Jul 10, 2018 by sqing
Source: Nguyen Duy Tung, Zhou Yuan Zhe
This is the "The Interesting Around Technical Analysis Three Variable Inequalities" article by Nguyen Duy Tung (Lionel Messi) and Zhou Yuan Zhe (red3), and as I promised, I'm posting it. As it's clear from the name of the article, it's about three variables inequalities.

Please write your comments and suggestions about this article which I've attached below.
3 replies
Amir Hossein
May 22, 2011
sqing
Jul 10, 2018
J
Technical Analysis of Three Variable Inequalities
G H J
Reveal topic tags
inequalities
articles
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Source: Nguyen Duy Tung, Zhou Yuan Zhe
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Amir Hossein
5452 posts
Amir Hossein
#1 h  May 22, 2011, 5:29 PM • 11 Y
Y by Pinphong, Rubaiya, Wizard_32, mathleticguyyy, hansduran0123, Adventure10, Mango247, and 4 other users
This is the "The Interesting Around Technical Analysis Three Variable Inequalities" article by Nguyen Duy Tung (Lionel Messi) and Zhou Yuan Zhe (red3), and as I promised, I'm posting it. As it's clear from the name of the article, it's about three variables inequalities.

Please write your comments and suggestions about this article which I've attached below.
Attachments:
The Interesting Around Technical Analysis Three Variable Inequalities - Nguyen Duy Tung, Zhou Yuan Zhe.pdf (304kb)
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Dr Sonnhard Graubner
16100 posts
Dr Sonnhard Graubner
#2 h  Feb 3, 2016, 6:57 PM • 3 Y
Y by 62861, me9hanics, Adventure10
Amir Hossein wrote:
This is the "The Interesting Around Technical Analysis Three Variable Inequalities" article by Nguyen Duy Tung (Lionel Messi) and Zhou Yuan Zhe (red3), and as I promised, I'm posting it. As it's clear from the name of the article, it's about three variables inequalities.

Please write your comments and suggestions about this article which I've attached below.

hello, your link is out of order!
Sonnhard.
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me9hanics
375 posts
me9hanics
#3 h  Jul 10, 2018, 12:55 PM • 2 Y
Y by Adventure10, Mango247
Yes the link is broken, I just came here from Olympiad Books :(
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sqing
41646 posts
sqing
#4 h  Jul 10, 2018, 1:35 PM • 4 Y
Y by Amir Hossein, Durjoy1729, Adventure10, Mango247
The Interesting Around Technical Analysis Three Variable Inequalities
This post has been edited 1 time. Last edited by sqing, Jul 10, 2018, 1:36 PM
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