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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
J
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A simple evaluation
jayme   0
a few seconds ago
Dear Mathlinkers,
a very simple question:

evaluate Arctan (-1 +√2) in degree

Sincerely
Jean-Louis
0 replies
jayme
a few seconds ago
0 replies
J
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cyc sum (a+1)\sqrt{2a(1-a)} \geq 8(ab+bc+ca)
Amir Hossein   11
N 3 minutes ago by Pseudo_Matter
Source: Greece JBMO TST 2017, Problem 1
Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that
$$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$Also, find the values of $a,b,c$ for which the equality happens.
11 replies
1 viewing
Amir Hossein
Jun 25, 2018
Pseudo_Matter
3 minutes ago
J
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inequalities proplem
Cobedangiu   1
N 5 minutes ago by sqing
$x,y\in R^+$ and $x+y-2\sqrt{x}-\sqrt{y}=0$. Find min A (and prove):
$A=\sqrt{\dfrac{5}{x+1}}+\dfrac{16}{5x^2y}$
1 reply
Cobedangiu
3 hours ago
sqing
5 minutes ago
J
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ARMO 2025
Oksutok   1
N 7 minutes ago by Oksutok
All-Russian MO 2025 Day 1
1 reply
Oksutok
5 hours ago
Oksutok
7 minutes ago
J
No more topics!
H
J
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Incircle cuts
chaotic_iak   1
N Mar 4, 2012 by jayme
Source: 2012 Indonesia Round 2 TST 2 Problem 2
Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.
1 reply
chaotic_iak
Mar 4, 2012
jayme
Mar 4, 2012
J
Incircle cuts
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Reveal topic tags
geometry
incenter
circumcircle
geometry unsolved
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G H BBookmark kLocked kLocked NReply
Source: 2012 Indonesia Round 2 TST 2 Problem 2
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chaotic_iak
2932 posts
chaotic_iak
#1 h  Mar 4, 2012, 11:33 AM • 2 Y
Y by Adventure10, Mango247
Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.
This post has been edited 1 time. Last edited by chaotic_iak, Mar 18, 2012, 2:32 PM
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The post below has been deleted. Click to close.
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jayme
9776 posts
jayme
#2 h  Mar 4, 2012, 12:17 PM • 1 Y
Y by Adventure10
Dear Mathlinkers,
see
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=329267

Sincerely
Jean-Louis
Z K Y
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