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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
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
Yesterday at 3:18 PM
0 replies
J
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hard problem
Cobedangiu   1
N 17 minutes ago by Cobedangiu
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
1 reply
+1 w
Cobedangiu
Yesterday at 6:11 PM
Cobedangiu
17 minutes ago
J
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(Original version) Same number of divisors
MNJ2357   2
N 36 minutes ago by john0512
Source: 2024 Korea Summer Program Practice Test P8 (original version)
For a positive integer \( n \), let \( \tau(n) \) denote the number of positive divisors of \( n \). Determine whether there exists a positive integer triple \( a, b, c \) such that there are exactly $1012$ positive integers \( K \) not greater than $2024$ that satisfies the following: the equation
\[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \]holds for some positive integers $x,y,z$.
2 replies
MNJ2357
Aug 12, 2024
john0512
36 minutes ago
J
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An inequality problem
Arithmetic_fighter   0
an hour ago
Given $a,b,c \in \mathbb R$ such that $a^2+b^2+c^2=3$. Prove that
$$\frac{a b}{c^2+a^2+1}+\frac{b c}{a^2+b^2+1}+\frac{c a}{b^2+c^2+1} \leq 1$$
0 replies
Arithmetic_fighter
an hour ago
0 replies
J
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Geometry :3c
popop614   3
N an hour ago by ItzsleepyXD
Source: MINE :<
Quadrilateral $ABCD$ has an incenter $I$ Suppose $AB > BC$. Let $M$ be the midpoint of $AC$. Suppose that $MI \perp BI$. $DI$ meets $(BDM)$ again at point $T$. Let points $P$ and $Q$ be such that $T$ is the midpoint of $MP$ and $I$ is the midpoint of $MQ$. Point $S$ lies on the plane such that $AMSQ$ is a parallelogram, and suppose the angle bisectors of $MCQ$ and $MSQ$ concur on $IM$.

The angle bisectors of $\angle PAQ$ and $\angle PCQ$ meet $PQ$ at $X$ and $Y$. Prove that $PX = QY$.
3 replies
popop614
4 hours ago
ItzsleepyXD
an hour ago
J
No more topics!
H
J
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A simple and nice inequality in a triangle.
Virgil Nicula   7
N Jun 26, 2016 by Virgil Nicula
Source: Own (?)
I don't meet this inequality until today ...

PP. Let $ABC$ be a triangle with the angled-bisectors $BE$ , $CF$ where $E\in AC$ , $F\in AB\ .$ Prove that $\boxed{\ EF^2\ \le\ BF\cdot CE\ }\ (*)\ .$
7 replies
Virgil Nicula
May 19, 2016
Virgil Nicula
Jun 26, 2016
J
A simple and nice inequality in a triangle.
G H J
Reveal topic tags
inequalities
geometry
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G H BBookmark kLocked kLocked NReply
Source: Own (?)
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Virgil Nicula
7054 posts
Virgil Nicula
#1 h  May 19, 2016, 4:52 PM • 2 Y
Y by Adventure10, Mango247
I don't meet this inequality until today ...

PP. Let $ABC$ be a triangle with the angled-bisectors $BE$ , $CF$ where $E\in AC$ , $F\in AB\ .$ Prove that $\boxed{\ EF^2\ \le\ BF\cdot CE\ }\ (*)\ .$
Z K Y
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arqady
30169 posts
arqady
#2 h  May 19, 2016, 7:33 PM • 2 Y
Y by Adventure10, Mango247
Indeed, it's $(b-c)^2a(a+b+c)\geq0$.
Z K Y
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aopser123
136 posts
aopser123
#3 h  May 19, 2016, 7:49 PM • 2 Y
Y by Adventure10, Mango247
arqady wrote:
Indeed, it's $(b-c)^2a(a+b+c)\geq0$.

How did you get this?
Z K Y
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Virgil Nicula
7054 posts
Virgil Nicula
#4 h  May 19, 2016, 8:38 PM • 2 Y
Y by Adventure10, Mango247
arqady wrote:
Indeed, it's $(b-c)^2a(a+b+c)\geq0$.

Yes, it is and my conclusion. See proposed problem P6 from here. Please, met you at least once this inequality ?! Thank you.
This post has been edited 2 times. Last edited by Virgil Nicula, May 20, 2016, 12:57 AM
Z K Y
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arqady
30169 posts
arqady
#5 h  May 19, 2016, 10:35 PM • 2 Y
Y by Adventure10, Mango247
No, I did not see it before.
Z K Y
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Virgil Nicula
7054 posts
Virgil Nicula
#6 h  May 19, 2016, 11:06 PM • 2 Y
Y by Adventure10, Mango247
Thank you. The your words say me that the previous inequality is own. Other interesting

inequality $:$ if $X$ , $Y$ are the projections of $E$ , $F$ on $BC\ ,$ then $XY\le \frac a2\cdot\left(\frac b{a+b}+\frac c{a+c}\right)\ .$
This post has been edited 2 times. Last edited by Virgil Nicula, Jun 26, 2016, 6:20 PM
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arqady
30169 posts
arqady
#9 h  Jun 26, 2016, 6:56 AM • 1 Y
Y by Adventure10
Are these projections to $BC$?
This post has been edited 1 time. Last edited by arqady, Jun 26, 2016, 6:56 AM
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Virgil Nicula
7054 posts
Virgil Nicula
#10 h  Jun 26, 2016, 6:24 PM • 1 Y
Y by Adventure10
Sorry! Yes, I edited now.
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