three variable

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

three variable

G H J

Reveal topic tags

inequalities unsolved

L

G H BBookmark kLocked kLocked NReply

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

121 posts

#1 h Jun 4, 2008, 3:41 PM • 2 Y

Y by Adventure10, Mango247

problem 1: let $a+b+c=3$

prove that $a^3b+b^3c+c^3a+6abc \le 9$

problem 2: let $a^2+b^2+c^2+abc=4$

prove that $a+b+c\ge \sqrt{abc}+2$

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

30182 posts

arqady

#2 h Jun 4, 2008, 8:04 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

bunhiacovski wrote:

problem 2: let $a^2 + b^2 + c^2 + abc = 4$

prove that $a + b + c\ge \sqrt {abc} + 2$

Try $a=b=-1$ and $c=1.$

By the way, for non-negative $a,$ $b$ and $c$ it's true:
If $c=0$ then $a+b=\sqrt{a^2+b^2+2ab}\geq\sqrt{a^2+b^2}=2.$
Thus we can assume that $a,$ $b$ and $c$ are positive numbers.
Let $a=2\sqrt{\frac{yz}{(x+y)(x+z)}},$ $b=2\sqrt{\frac{xz}{(x+y)(y+z)}}$ and $c=2\sqrt{\frac{xy}{(x+z)(y+z)}},$
where $x,$ $y$ and $z$ are positive numbers.
Hence, we need to prove that:
$\sum_{cyc}2\sqrt{\frac{xy}{(x+z)(y+z)}}\geq\frac{8xyz}{(x+y)(x+z)(y+z)}+2,$ which is equivalent to
$\sum_{cyc}(x+y)\sqrt{xy(x+z)(y+z)}\geq\sum_{cyc}(x^2y+x^2z+2xyz).$
But $\sqrt{(x+z)(y+z)}\geq\sqrt{xy}+z.$
Id est, it remains to prove that $\sum_{cyc}(x+y)(xy+z\sqrt{xy})\geq\sum_{cyc}(x^2y+x^2z+2xyz),$
which is equivalent to $\sum_{cyc}z\sqrt{xy}(\sqrt x-\sqrt y)^2\geq0.$

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

121 posts

#3 h Jun 15, 2008, 9:15 AM • 1 Y

Y by Adventure10

problem 2, who can bureau?

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

30182 posts

arqady

#4 h Jan 27, 2013, 6:17 AM • 1 Y

Y by Adventure10

arqady wrote:

Hence, we need to prove that:
$\sum_{cyc}2\sqrt{\frac{xy}{(x+z)(y+z)}}\geq\frac{8xyz}{(x+y)(x+z)(y+z)}+2$

It should be $\sum_{cyc}2\sqrt{\frac{xy}{(x+z)(y+z)}}\geq\sqrt{\frac{8xyz}{(x+y)(x+z)(y+z)}}+2$ ,

which is true (after squaring of the both sides and using C-S).

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

30182 posts

arqady

#5 h Apr 12, 2016, 4:38 AM • 4 Y

Y by luofangxiang, mudok, Adventure10, Mango247

I think I meant the following reasoning.
We need to prove that $\sum_{cyc}\sqrt{xy(x+y)}\geq\sqrt{2xyz}+\sqrt{(x+y)(x+z)(y+z)}$ , which after squaring of the both sides gives
$\sum_{cyc}\sqrt{x(x+y)(x+z)}\geq2\sqrt{xyz}+\sqrt{2(x+y)(x+z)(y+z)}$ .
Since by C-S and Schur
$\sum_{cyc}\sqrt{x(x+y)(x+z)}=\sqrt{\sum_{cyc}\left(x^3+x^2y+x^2z+xyz+2\sqrt{(x^2(x+y+z)+xyz)(y^2(x+y+z)+xyz)}\right)}\geq$
$\geq\sqrt{\sum_{cyc}\left(x^3+x^2y+x^2z+xyz+2(xy(x+y+z)+xyz)\right)}=\sqrt{\sum_{cyc}(x^3+3x^2y+3x^2z+5xyz)}\geq$
$\geq2\sqrt{(x+y+z)(xy+xz+yz)}$ , it remains to prove that
$2\sqrt{(x+y+z)(xy+xz+yz)}\geq2\sqrt{xyz}+\sqrt{2(x+y)(x+z)(y+z)}$ .
Let $\sum_{cyc}(x^2+x^2z)=6kxyz$ . Hence, $k\geq1$ and we need to prove that
$2\sqrt{6k+3}\geq2+\sqrt{12k+4}$ , which is obvious.

This post has been edited 2 times. Last edited by arqady, Apr 12, 2016, 6:48 AM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

3377 posts

mudok

#6 h Apr 12, 2016, 4:57 AM • 3 Y

Y by arqady, Adventure10, Mango247

arqady wrote:

We need to prove that $\sum_{cyc}xy\sqrt{x+y}\geq\sqrt{2xyz}+\sqrt{(x+y)(x+z)(y+z)}$

Typo . Must be $\sum_{cyc}\sqrt{xy(x+y)}$

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

4613 posts

#7 h Apr 12, 2016, 5:19 AM • 2 Y

Y by Adventure10, Mango247

//cdn.artofproblemsolving.com/images/6/5/e/65e5d337900a2d02bc64bcb8dc8ddcd1df0e919c.jpg

This post has been edited 1 time. Last edited by luofangxiang, Apr 12, 2016, 5:22 AM
Reason: 123

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a