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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
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0 replies
1 viewing
jlacosta
Yesterday at 3:18 PM
0 replies
Inspired by old results
sqing   2
N 5 minutes ago by sqing
Source: Own
Let $ a,b,c \ge  \frac{1}{21} $ and $ a+b+c=1. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq\frac{1}{729} $$Let $ a,b,c \ge  \frac{1}{10} $ and $ a+b+c=2. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq\frac{64}{729} $$Let $ a,b,c \ge  \frac{1}{11} $ and $ a+b+c=2. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq \frac{145161}{1771561} $$
2 replies
1 viewing
sqing
an hour ago
sqing
5 minutes ago
Very interesting inequalities
sqing   2
N 13 minutes ago by sqing
Source: Own
Let $  x ,y \geq 0 $ and $ x^2 -x+ \frac{1}{2}y\leq 1.$ Prove that
$$x^2 + ky  \leq \frac{k(5k-2)}{2k-1}$$Where $ k\in N^+.$
$$x^2 + y  \leq 3$$$$x^2 +  2y  \leq \frac{16}{3}$$
2 replies
sqing
2 hours ago
sqing
13 minutes ago
high school maths
aothatday   0
19 minutes ago
Source: my creation
find $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:
$(x-y)(f(x)+f(y)) \leq f(x^2-y^2)$
0 replies
aothatday
19 minutes ago
0 replies
n x n square and strawberries
pohoatza   18
N 44 minutes ago by shanelin-sigma
Source: IMO Shortlist 2006, Combinatorics 4, AIMO 2007, TST 4, P2
A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$.

Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows:

A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.
18 replies
+1 w
pohoatza
Jun 28, 2007
shanelin-sigma
44 minutes ago
No more topics!
A simple inequality for highschool students
can_hang2007   37
N Nov 12, 2021 by sqing
Let $a ,\; b,\; c$ be positive real numbers such that
$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$
Prove that
$b^2+3 \ge 2b(a+c).$
37 replies
can_hang2007
Nov 3, 2010
sqing
Nov 12, 2021
A simple inequality for highschool students
G H J
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can_hang2007
2948 posts
#1 • 7 Y
Y by xyzz, ZHEKSHEN, a_friendwr_a, son7, Ha_ha_ha, Adventure10, Mango247
Let $a ,\; b,\; c$ be positive real numbers such that
$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$
Prove that
$b^2+3 \ge 2b(a+c).$
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sqing
41401 posts
#2 • 4 Y
Y by a_friendwr_a, Ha_ha_ha, Mango247, Mango247
can_hang2007 wrote:
Let $a ,\; b,\; c$ be positive real numbers such that $\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$ Prove that
$$b^2+3 \ge 2b(a+c).$$
Good.
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ZHEKSHEN
25 posts
#3 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
By Titu: 3>= (a^2) + (b^2) + (c^2)
Now we can rewrite inequality:
2(b^2)+ (a^2) + (c^2)>= 2ab+ 2bc
By AM-GM:
(b^2)+(a^2)>= 2ab
(b^2)+(c^2)>= 2bc
Summing up two inequality, we have done
P.S: sorry for latex
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menpo
209 posts
#4 • 4 Y
Y by a_friendwr_a, Ha_ha_ha, Mango247, Mango247
ZHEKSHEN wrote:
By Titu: 3>= (a^2) + (b^2) + (c^2)
Now we can rewrite inequality:
2(b^2)+ (a^2) + (c^2)>= 2ab+ 2bc
By AM-GM:
(b^2)+(a^2)>= 2ab
(b^2)+(c^2)>= 2bc
Summing up two inequality, we have done
P.S: sorry for latex

Are you sure that $3\ge a^2+b^2+c^2$ ?
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AleksaS
41 posts
#5 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
Adilet160205 wrote:
ZHEKSHEN wrote:
By Titu: 3>= (a^2) + (b^2) + (c^2)
Now we can rewrite inequality:
2(b^2)+ (a^2) + (c^2)>= 2ab+ 2bc
By AM-GM:
(b^2)+(a^2)>= 2ab
(b^2)+(c^2)>= 2bc
Summing up two inequality, we have done
P.S: sorry for latex

Are you sure that $3\ge a^2+b^2+c^2$ ?
By Titu you get that $9\ge 2(a^2 + b^2 + c^2) + 3$ which means that $3 \ge a^2 + b^2 + c^2$ and by $AM - GM$ you get that $3 \ge 2ab + 2bc - b^2$ from which you get needed inequality.
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pi_quadrat_sechstel
583 posts
#6 • 3 Y
Y by xyzz, a_friendwr_a, Ha_ha_ha
AleksaS wrote:
Adilet160205 wrote:
ZHEKSHEN wrote:
By Titu: 3>= (a^2) + (b^2) + (c^2)
Now we can rewrite inequality:
2(b^2)+ (a^2) + (c^2)>= 2ab+ 2bc
By AM-GM:
(b^2)+(a^2)>= 2ab
(b^2)+(c^2)>= 2bc
Summing up two inequality, we have done
P.S: sorry for latex

Are you sure that $3\ge a^2+b^2+c^2$ ?
By Titu you get that $9\ge 2(a^2 + b^2 + c^2) + 3$ which means that $3 \ge a^2 + b^2 + c^2$ and by $AM - GM$ you get that $3 \ge 2ab + 2bc - b^2$ from which you get needed inequality.

No.

By Titu you get
\[
\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}\geq\frac{9}{2(a^2+b^2+c^2)+3}
\]and not nessecarily
\[
\frac{9}{2(a^2+b^2+c^2)+3}\geq1
\]
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AleksaS
41 posts
#7 • 5 Y
Y by a_friendwr_a, Ha_ha_ha, Mango247, Mango247, Mango247
In this case it would be useful if we can prove that but I don't see a way to do that.
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pi_quadrat_sechstel
583 posts
#8 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
AleksaS wrote:
In this case it would be useful if we can prove that but I don't see a way to do that.

It is not true. For distinct $a,b,c$ with $1=\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}$, we get by Titu
\[
1=\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}>\frac{9}{2(a^2+b^2+c^2)+3}
\]
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alinazarboland
168 posts
#9 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
wrong solution
This post has been edited 1 time. Last edited by alinazarboland, Jan 8, 2021, 10:23 AM
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pi_quadrat_sechstel
583 posts
#10 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
alinazarboland wrote:
can_hang2007 wrote:
Let $a ,\; b,\; c$ be positive real numbers such that
$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$
Prove that
$b^2+3 \ge 2b(a+c).$

Let's make $a ,b$ fix. Larger $c$ makes condition stronger and stronger claim .so we can assume that

$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} =1$

now by T2 lemma we have $a^2 + b^2 +c^2 <3$ the rest is easy...

No, we have $a^2+b^2+c^2\geq3$
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arqady
30171 posts
#11 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
alinazarboland wrote:

now by T2 lemma we have $a^2 + b^2 +c^2 <3$ the rest is easy...
Try $c\rightarrow0^+,$ $a=2$ and $b=\sqrt{\frac{\sqrt{89}-7}{4}}.$
This post has been edited 1 time. Last edited by arqady, Jan 8, 2021, 10:10 AM
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csav10
382 posts
#12 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
It is enough to show that
$b^2+3 \ge 2b\sqrt{2(a^2+c^2)},$
which is not hard.
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Universes
96 posts
#13 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
csav10 wrote:
It is enough to show that
$b^2+3 \ge 2b\sqrt{2(a^2+c^2)},$
which is not hard.

Are you sure?
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csav10
382 posts
#14 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
I am sure that $b^2+3 \ge 2b\sqrt{2(a^2+c^2)}$ is wrong, but $\sqrt{2(b^2+1)}\ge b(a+c)$ is true.
This post has been edited 1 time. Last edited by csav10, Jan 11, 2021, 6:30 AM
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pi_quadrat_sechstel
583 posts
#15 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
We need to prove
\[
2\left(a^2+b^2+c^2\right)+3\geq(a+b+c)^2+(a-c)^2
\]Thus we can assume w.l.o.g $a\geq b\geq c$. It suffices to prove
\[
\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}<1
\]for reals $a\geq b\geq c>0$ with $3<2b(a+c)-b^2$. Replace $(a,b,c)$ with
\[
\left(a\cdot\sqrt{\frac{3}{2b(a+c)-b^2}},b\cdot\sqrt{\frac{3}{2b(a+c)-b^2}},c\cdot\sqrt{\frac{3}{2b(a+c)-b^2}}\right)
\]and note that $\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}$ rises. It suffices to prove
\[
\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}\leq1
\]for reals $a\geq b\geq c>0$ with $3=2b(a+c)-b^2$. Write the inequality in the homogenous from
\[
\frac{1}{3c^2+2b^2+2b(a+c)}+\frac{1}{3a^2+2b^2+2b(a+c)}+\frac{1}{3a^2+3c^2+2b(a+c)-b^2}\leq\frac{1}{2b(a+c)-b^2}
\]Define $x,y\geq0$ by $b=c+x,a=b+y$. The inequality is eqiuvalent to
\begin{align*}
36c^4(x-y)^2+6c^3x(x-y)^2+54c^4x^2+54c^4y^2+274c^3x^3+132c^3xy^2+80c^3y^3\\
+330c^2x^4+228c^2x^3y+42c^2x^2y^2+72c^2xy^3+30c^2y^4+180cx^5+255cxy^4\\
+84cx^3y^2+12cx^2y^3+6cxy^4+40x^6+90x^5y+63x^4y^2+18x^3y^3+3x^2y^4\geq0
\end{align*}which is obviously true.
This post has been edited 1 time. Last edited by pi_quadrat_sechstel, Jan 8, 2021, 9:45 PM
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Universes
96 posts
#16 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
You can't order the variables.
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pi_quadrat_sechstel
583 posts
#17 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
Universes wrote:
You can't order the variables.

I can. Since $b^2+3 \ge 2b(a+c)\Leftrightarrow2\left(a^2+b^2+c^2\right)+3\geq(a+b+c)^2+(a-c)^2$ the $RHS$ is maximal for $a\geq b\geq c$.
This post has been edited 1 time. Last edited by pi_quadrat_sechstel, Jan 9, 2021, 9:49 AM
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pi_quadrat_sechstel
583 posts
#18 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
can_hang2007 wrote:
Let $a ,\; b,\; c$ be positive real numbers such that
$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$
Prove that
$b^2+3 \ge 2b(a+c).$

Prove that $\lambda=1$ is the best constant with
\[
2(a^2+b^2+c^2)+3\geq(a+b+c)^2+\lambda(a-c)^2
\]for all nonnegative reals $a,b,c$ with
\[
\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1
\]
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Universes
96 posts
#19 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
pi_quadrat_sechstel wrote:
Universes wrote:
You can't order the variables.

I can. Since $b^2+3 \ge 2b(a+c)\Leftrightarrow2\left(a^2+b^2+c^2\right)+3\geq(a+b+c)^2+(a-c)^2$ the $LHS$ is maximal for $a\geq b\geq c$.

LHS or RHS?
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pi_quadrat_sechstel
583 posts
#21 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
RHS. I've corrected it now.
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Universes
96 posts
#22 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
pi_quadrat_sechstel wrote:
pi_quadrat_sechstel wrote:
Universes wrote:
You can't order the variables.

I can. Since $b^2+3 \ge 2b(a+c)\Leftrightarrow2\left(a^2+b^2+c^2\right)+3\geq(a+b+c)^2+(a-c)^2$ the $RHS$ is maximal for $a\geq b\geq c$.

So , also you have to consider the case $c\ge b\ge a. $
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pi_quadrat_sechstel
583 posts
#25 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
Universes wrote:

So , also you have to consider the case $c\ge b\ge a. $

No, we can switch $a,c$.
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Universes
96 posts
#26 • 3 Y
Y by a_friendwr_a, Ha_ha_ha, Mango247
Your proof is equivalent to knowing the condition $b^2+3 \ge 2b (a+c) $ and proving the inequality \[
\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}\leq1
\].
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pi_quadrat_sechstel
583 posts
#27 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
Universes wrote:
Your proof is equivalent to knowing the condition $b^2+3 \ge 2b (a+c) $ and proving the inequality \[
\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}\leq1
\].

It proves $b^2+3<2b(a+c)\Rightarrow\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}<1$. Since $(\lnot B\Rightarrow\lnot A)\Leftrightarrow(A\Rightarrow B)$ it proves $\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}\geq1\Rightarrow b^2+3\geq2b(a+c)$, too.
This post has been edited 1 time. Last edited by pi_quadrat_sechstel, Jan 9, 2021, 10:19 AM
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csav10
382 posts
#28 • 2 Y
Y by a_friendwr_a, Ha_ha_ha
The equality occurs for $a=b=c=1$.
This post has been edited 1 time. Last edited by csav10, Jan 11, 2021, 6:32 AM
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math90
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#29 • 7 Y
Y by arqady, a_friendwr_a, xyzz, hangb6pbc, dragonheart6, mudok, Ha_ha_ha
can_hang2007 wrote:
Let $a ,\; b,\; c$ be positive real numbers such that
$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$
Prove that
$b^2+3 \ge 2b(a+c).$

After expanding, the condition is equivalent to
\[2\left(a^2+b^2+c^2+1\right)\ge\prod\left(a^2+b^2\right)\]Details

Equivalently, we have
\[2+\frac{2\left(b^2+1\right)}{a^2+c^2}\ge\left(a^2+b^2\right)\left(b^2+c^2\right)\]Since $2\left(a^2+c^2\right)\ge \left(a+c\right)^2$, we have
\[2+\frac{4\left(b^2+1\right)}{\left(a+c\right)^2}\ge 2+\frac{2\left(b^2+1\right)}{a^2+c^2}\]Moreover, by C-S inequality we have
\[\left(a^2+b^2\right)\left(b^2+c^2\right)\ge b^2\left(a+c\right)^2\]Hence,
\begin{align*}
&2+\frac{4\left(b^2+1\right)}{\left(a+c\right)^2}\ge b^2\left(a+c\right)^2\\
&\implies 2\left(a+c\right)^2+4\ge b^2\left(a+c\right)^4-4b^2\\
&\implies 2\left[\left(a+c\right)^2+2\right]\ge b^2\left[\left(a+c\right)^2-2\right]\left[\left(a+c\right)^2+2\right]\\
&\implies 2\ge b^2\left[\left(a+c\right)^2-2\right]\\
&\implies b^2\left(a+c\right)^2\le 2\left(b^2+1\right)\le\frac{\left(b^2+1+2\right)^2}{4}=\frac{\left(b^2+3\right)^2}{4}\\
&\implies 2b\left(a+c\right)\le b^2+3
\end{align*}as desired.
This post has been edited 1 time. Last edited by math90, Jul 30, 2021, 12:10 PM
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arqady
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#30 • 3 Y
Y by a_friendwr_a, math90, Ha_ha_ha
Beautiful proof! :-D
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csav10
382 posts
#31 • 1 Y
Y by Ha_ha_ha
arqady wrote:
Beautiful proof! :-D
I agree with you. :flex:
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pi_quadrat_sechstel
583 posts
#32 • 2 Y
Y by Ha_ha_ha, Mango247
pi_quadrat_sechstel wrote:
can_hang2007 wrote:
Let $a ,\; b,\; c$ be positive real numbers such that
$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$
Prove that
$b^2+3 \ge 2b(a+c).$

Prove that $\lambda=1$ is the best constant with
\[
2(a^2+b^2+c^2)+3\geq(a+b+c)^2+\lambda(a-c)^2
\]for all nonnegative reals $a,b,c$ with
\[
\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1
\]

Bump.
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mudok
3377 posts
#33 • 3 Y
Y by dragonheart6, Ha_ha_ha, xyzz
can_hang2007 wrote:
Let $a ,\; b,\; c$ be positive real numbers such that
$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$
Prove that
$b^2+3 \ge 2b(a+c).$

$\sum_{cyc}\frac{1}{a^2+b^2+1} \ge 1 \iff 4\ge \sum_{cyc} \frac{(a+b)^2}{a^2+b^2+1} + \sum_{cyc} \frac{(a-b)^2}{a^2+b^2+1} $

By Cauchy-Schwarz:

$ \sum_{cyc} \frac{(a+b)^2}{a^2+b^2+1}\ge  \frac{4(a+b+c)^2}{2(a^2+b^2+c^2)+3}$

$ \sum_{cyc} \frac{(a-b)^2}{a^2+b^2+1}\ge \frac{4(a-c)^2}{2(a^2+b^2+c^2)+3}$

So, we have:

$4\ge \frac{4(a+b+c)^2+4(a-c)^2}{2(a^2+b^2+c^2)+3} \iff b^2+3 \ge 2b(a+c).$
This post has been edited 4 times. Last edited by mudok, Jan 11, 2021, 5:59 AM
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sqing
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#34 • 1 Y
Y by Ha_ha_ha
Beautiful .
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csav10
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#35 • 2 Y
Y by Ha_ha_ha, Mango247
Actually, the stronger inequality holds:
$$\sqrt{2(b^2+1)}\ge b(a+c).$$
pi_quadrat_sechstel wrote:

Prove that $\lambda=1$ is the best constant with
\[
2(a^2+b^2+c^2)+3\geq(a+b+c)^2+\lambda(a-c)^2
\]for all nonnegative reals $a,b,c$ with
\[
\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1
\]
The hypothesis
$$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1$$is satisfied for $b=c=0$ and all $a\ge 0$, when the inequality
$$2(a^2+b^2+c^2)+3\geq(a+b+c)^2+\lambda(a-c)^2$$becomes
$$3\ge( \lambda -1)a^2.$$Clearly, this is true for any $a\ge 0$ if and only if $\lambda\le 1$.
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mudok
3377 posts
#36 • 3 Y
Y by Ha_ha_ha, Mango247, Mango247
csav10 wrote:
Actually, the stronger inequality holds:
$$\sqrt{2(b^2+1)}\ge b(a+c).$$

You can see its proof in math90's proof. See post #29
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sqing
41401 posts
#37 • 1 Y
Y by Ha_ha_ha
can_hang2007 wrote:
Let $a ,\; b,\; c$ be positive real numbers such that $\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$ Prove that
$$b^2+3 \ge 2b(a+c).$$
Solution of Zhangyanzong:
$$\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1\iff  2\geq \frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1}+\frac{a^2+b^2}{a^2+b^2+1} $$$$\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1}+\frac{a^2+b^2}{a^2+b^2+1} \geq \frac{(\sqrt{b^2+c^2}+\sqrt{c^2+a^2}+\sqrt{a^2+b^2})^2}{2(a^2+b^2+c^2)+3}$$$$2\geq \frac{(\sqrt{b^2+c^2}+\sqrt{c^2+a^2}+\sqrt{a^2+b^2})^2}{2(a^2+b^2+c^2)+3}$$$$b^2+3\geq \sqrt{(a^2+b^2)(a^2+c^2)}+\sqrt{(c^2+a^2)(c^2+b^2)}+\sqrt{(a^2+b^2)(b^2+c^2)}-a^2-c^2$$$$\geq a^2+bc+c^2+ab+ab+bc-a^2-c^2=2b(a+c).$$
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sqing
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#38
Y by
can_hang2007 wrote:
Let $a ,\; b,\; c$ be positive real numbers such that $\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1} \ge 1.$ Prove that
$$b^2+3 \ge 2b(a+c).$$
Maybe inspired by here

Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a+b+c=3$ . Prove that $$ \frac{1}{2+a^{2}+b^{2}}+\frac{1}{2+b^{2}+c^{2}}+\frac{1}{2+c^{2}+a^{2}} \leq \frac{3}{4}$$Iran TST 2009-Day1-P3
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sqing
41401 posts
#39 • 3 Y
Y by Mango247, Mango247, Mango247
Let $a,b,c$ be positive real numbers such $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1} \geq 1$.Prove that$$a^2b^2+b^2c^2+c^2a^2 \leq 3.$$Let $a,b,c$ be positive real numbers such $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1} \geq 1$.Prove that$$ab+bc+ca \leq 3.$$Solution:
$$\frac{2+c^2}{(a+b+c)^2}\geq \frac{1+1+c^2}{(a^2+b^2+1)(1+1+c^2)}=\frac{1}{(a^2+b^2+1)}$$$$6+a^2+b^2+c^2\geq (a+b+c)^2$$$$ab+bc+ca \leq 3$$h
This post has been edited 2 times. Last edited by sqing, Nov 5, 2021, 10:24 AM
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#40
Y by
Let $a ,b,c$ be positive real numbers such that $\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}+\frac{1}{a^2+b^2+1}\geq 1.$ Prove that
$$b^2+3 \ge b\sqrt {2(a^2+c^2)}$$
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sqing
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#41
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Let $a,b,c$ be positive real numbers such that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}\geq 1.$ Prove that$$c^k+\frac{3}{c^k}\geq 2(a^k+b^k)$$Where $k=1,2.$
Let $a,b,c$ be positive real numbers such that $\frac{a+b}{a^2+b^2+1}+\frac{b+c}{b^2+c^2+1}+\frac{c+a}{c^2+a^2+1}\geq 2.$ Prove that
$$c^k+\frac{3}{c^k}\geq 2(a^k+b^k)$$Where $k=1,2,3,4.$
Let $a,b,c$ be positive real numbers such that $\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}\leq 1.$ Prove that$$c+\frac{3}{c}\geq 2(a+b)$$
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