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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
J
H
A simple evaluation
jayme   0
a minute ago
Dear Mathlinkers,
a very simple question:

evaluate Arctan (-1 +√2) in degree

Sincerely
Jean-Louis
0 replies
jayme
a minute ago
0 replies
J
H
cyc sum (a+1)\sqrt{2a(1-a)} \geq 8(ab+bc+ca)
Amir Hossein   11
N 4 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
4 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
H
Quadrilateral in a quadrilateral
jgnr   2
N Jun 29, 2008 by jgnr
Source: Singapore 2008 Junior 2nd Round
In the quadrilateral $ PQRS$, $ A$, $ B$, $ C$ and $ D$ are the midpoints of the sides $ PQ$, $ QR$, $ RS$ and $ SP$ respectively, and $ M$ is the midpoint of $ CD$. Suppose $ H$ is the point on the line $ AM$ such that $ HC=BC$. Prove that $ \angle BHM=90^\circ$.
2 replies
jgnr
Jun 28, 2008
jgnr
Jun 29, 2008
J
Quadrilateral in a quadrilateral
G H J
Reveal topic tags
geometry
parallelogram
analytic geometry
geometry unsolved
L
G H BBookmark kLocked kLocked NReply
Source: Singapore 2008 Junior 2nd Round
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jgnr
1343 posts
jgnr
#1 h  Jun 28, 2008, 11:26 PM • 2 Y
Y by Adventure10, Mango247
In the quadrilateral $ PQRS$, $ A$, $ B$, $ C$ and $ D$ are the midpoints of the sides $ PQ$, $ QR$, $ RS$ and $ SP$ respectively, and $ M$ is the midpoint of $ CD$. Suppose $ H$ is the point on the line $ AM$ such that $ HC=BC$. Prove that $ \angle BHM=90^\circ$.
Z K Y
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plane geometry
467 posts
plane geometry
#2 h  Jun 29, 2008, 3:24 AM • 2 Y
Y by Adventure10, Mango247
easy to know ABCD is a parallelogram
CN=CH=BC=AD triangle BHN is a right triangle
done！
Attachments:
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jgnr
1343 posts
jgnr
#3 h  Jun 29, 2008, 3:37 AM • 2 Y
Y by Adventure10, Mango247
WOW!!! Great solution!

During the contest, I used coordinate geometry. :oops: But I didn't finish my solution. (I started in the last 3 minutes.)
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