Significance of assumption in competition inequality questions
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Refer to the problem below (IMO 2009 Shortlist)
Let $a, b, c$ be positive real numbers such that $$frac{1}{a} + frac{1}{b} + frac{1}{c} = a + b+ c$$
Prove that
$$frac{1}{(2a+b+c)^2} + frac{1}{(2a+b+c)^2} + frac{1}{(2a+b+c)^2} leqslant frac{3}{16} $$
In the solution given by the official short list solution book (Pg 16), it states that
Without loss of generality, we choose $$a +b+c = 1$$.
Thus, the problem becomes
$$ frac{1}{(1 + a)^2}
+ frac{1}{(1 + b)^2}
+frac{1}{(1 + b)^2}
leqslant frac{3}{16}(frac{1}{a} + frac{1}{b} + frac{1}{c})$$
Applying Jensen’s inequality to the function $$f(x) = frac{x}{ (1+ x)^2}$$
,which is concave for $0 ≤ x ≤ 2$
and increasing for $0 ≤ x ≤ 1$, we obtain
$$α
frac{a}{(1 + a)^2} + β
frac{b}{(1 + b)^2} + γ
frac{c}{(1 + c)^2} leqslant
(α + β + γ)
frac{A}{(1 + A)^2}$$
, where $A =frac{αa + βb + γc}{α + β + γ}.$
Choosing $α =
frac{1}{a}
, β =frac{1}{b},$ and $γ = frac{1}{c}$
, we can apply the harmonic-arithmetic-mean inequality
$$A =frac{3}{frac{1}{a} + frac{1}{b} + frac{1}{c}}
≤
frac{a + b + c
}{3}
=
frac{1}{3}
< 1$$
Finally we prove:
$$ frac{1}{(1 + a)^2}
+ frac{1}{(1 + b)^2}
+frac{1}{(1 + b)^2}
leqslant ( frac{1}{a} + frac{1}{b} + frac{1}{c}) frac{A}{(1 + A)^2} leqslant ( frac{1}{a} + frac{1}{b} + frac{1}{c})frac{A}{(1 + frac{1}{3})^2} = frac{3}{16}(frac{1}{a} + frac{1}{b} + frac{1}{c}) $$
However, Evan Chen's solution (Pg 4) differs by having $a +b+c=3$ and setting $f(x) = frac{1}{16x} - frac{1}{(x+3)^2}$ and allowing Jensen to resolve the rest.
The questions are as follows;
How do you choose before hand which $a + b+ c =$ to choose?
Why particularly does the assumption differ between the 2 solutions?
How does one know before hand the choosing of $α =
frac{1}{a}
, β =frac{1}{b},$ and $γ = frac{1}{c}$?Where and how did $A =frac{αa + βb + γc}{α + β + γ}$ come from?
Any help would be much appreciated.
inequality contest-math
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up vote
1
down vote
favorite
Refer to the problem below (IMO 2009 Shortlist)
Let $a, b, c$ be positive real numbers such that $$frac{1}{a} + frac{1}{b} + frac{1}{c} = a + b+ c$$
Prove that
$$frac{1}{(2a+b+c)^2} + frac{1}{(2a+b+c)^2} + frac{1}{(2a+b+c)^2} leqslant frac{3}{16} $$
In the solution given by the official short list solution book (Pg 16), it states that
Without loss of generality, we choose $$a +b+c = 1$$.
Thus, the problem becomes
$$ frac{1}{(1 + a)^2}
+ frac{1}{(1 + b)^2}
+frac{1}{(1 + b)^2}
leqslant frac{3}{16}(frac{1}{a} + frac{1}{b} + frac{1}{c})$$
Applying Jensen’s inequality to the function $$f(x) = frac{x}{ (1+ x)^2}$$
,which is concave for $0 ≤ x ≤ 2$
and increasing for $0 ≤ x ≤ 1$, we obtain
$$α
frac{a}{(1 + a)^2} + β
frac{b}{(1 + b)^2} + γ
frac{c}{(1 + c)^2} leqslant
(α + β + γ)
frac{A}{(1 + A)^2}$$
, where $A =frac{αa + βb + γc}{α + β + γ}.$
Choosing $α =
frac{1}{a}
, β =frac{1}{b},$ and $γ = frac{1}{c}$
, we can apply the harmonic-arithmetic-mean inequality
$$A =frac{3}{frac{1}{a} + frac{1}{b} + frac{1}{c}}
≤
frac{a + b + c
}{3}
=
frac{1}{3}
< 1$$
Finally we prove:
$$ frac{1}{(1 + a)^2}
+ frac{1}{(1 + b)^2}
+frac{1}{(1 + b)^2}
leqslant ( frac{1}{a} + frac{1}{b} + frac{1}{c}) frac{A}{(1 + A)^2} leqslant ( frac{1}{a} + frac{1}{b} + frac{1}{c})frac{A}{(1 + frac{1}{3})^2} = frac{3}{16}(frac{1}{a} + frac{1}{b} + frac{1}{c}) $$
However, Evan Chen's solution (Pg 4) differs by having $a +b+c=3$ and setting $f(x) = frac{1}{16x} - frac{1}{(x+3)^2}$ and allowing Jensen to resolve the rest.
The questions are as follows;
How do you choose before hand which $a + b+ c =$ to choose?
Why particularly does the assumption differ between the 2 solutions?
How does one know before hand the choosing of $α =
frac{1}{a}
, β =frac{1}{b},$ and $γ = frac{1}{c}$?Where and how did $A =frac{αa + βb + γc}{α + β + γ}$ come from?
Any help would be much appreciated.
inequality contest-math
I have found another solution. If you want I am ready to post it.
– Michael Rozenberg
Nov 15 at 20:03
Here is a problem for me, since $$(a+b+c)({1over a}+{1over b}+{1over c})geq 9$$ we have $a+b+cgeq 3$, so I don't understand either solution.
– greedoid
Nov 15 at 20:19
@MichaelRozenberg Can you explain please this phenomena?
– greedoid
Nov 15 at 20:23
@greedoid After homogenization we need to prove that $sumlimits_{cyc}frac{1}{(2a+b+c)^2}leqfrac{3}{16(a+b+c)}sumlimits_{cyc}frac{1}{a}$ and we can assume that even $a+b+c=17$, but we saw that the assuming was $a+b+c=1.$ In your example just $sumlimits_{cyc}frac{1}{a}geq9.$
– Michael Rozenberg
Nov 15 at 20:35
@greedoid See please the first inequality in my previous post. This inequality is equivalent to the given inequality, but it's homogeneous already.
– Michael Rozenberg
Nov 15 at 20:40
|
show 6 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Refer to the problem below (IMO 2009 Shortlist)
Let $a, b, c$ be positive real numbers such that $$frac{1}{a} + frac{1}{b} + frac{1}{c} = a + b+ c$$
Prove that
$$frac{1}{(2a+b+c)^2} + frac{1}{(2a+b+c)^2} + frac{1}{(2a+b+c)^2} leqslant frac{3}{16} $$
In the solution given by the official short list solution book (Pg 16), it states that
Without loss of generality, we choose $$a +b+c = 1$$.
Thus, the problem becomes
$$ frac{1}{(1 + a)^2}
+ frac{1}{(1 + b)^2}
+frac{1}{(1 + b)^2}
leqslant frac{3}{16}(frac{1}{a} + frac{1}{b} + frac{1}{c})$$
Applying Jensen’s inequality to the function $$f(x) = frac{x}{ (1+ x)^2}$$
,which is concave for $0 ≤ x ≤ 2$
and increasing for $0 ≤ x ≤ 1$, we obtain
$$α
frac{a}{(1 + a)^2} + β
frac{b}{(1 + b)^2} + γ
frac{c}{(1 + c)^2} leqslant
(α + β + γ)
frac{A}{(1 + A)^2}$$
, where $A =frac{αa + βb + γc}{α + β + γ}.$
Choosing $α =
frac{1}{a}
, β =frac{1}{b},$ and $γ = frac{1}{c}$
, we can apply the harmonic-arithmetic-mean inequality
$$A =frac{3}{frac{1}{a} + frac{1}{b} + frac{1}{c}}
≤
frac{a + b + c
}{3}
=
frac{1}{3}
< 1$$
Finally we prove:
$$ frac{1}{(1 + a)^2}
+ frac{1}{(1 + b)^2}
+frac{1}{(1 + b)^2}
leqslant ( frac{1}{a} + frac{1}{b} + frac{1}{c}) frac{A}{(1 + A)^2} leqslant ( frac{1}{a} + frac{1}{b} + frac{1}{c})frac{A}{(1 + frac{1}{3})^2} = frac{3}{16}(frac{1}{a} + frac{1}{b} + frac{1}{c}) $$
However, Evan Chen's solution (Pg 4) differs by having $a +b+c=3$ and setting $f(x) = frac{1}{16x} - frac{1}{(x+3)^2}$ and allowing Jensen to resolve the rest.
The questions are as follows;
How do you choose before hand which $a + b+ c =$ to choose?
Why particularly does the assumption differ between the 2 solutions?
How does one know before hand the choosing of $α =
frac{1}{a}
, β =frac{1}{b},$ and $γ = frac{1}{c}$?Where and how did $A =frac{αa + βb + γc}{α + β + γ}$ come from?
Any help would be much appreciated.
inequality contest-math
Refer to the problem below (IMO 2009 Shortlist)
Let $a, b, c$ be positive real numbers such that $$frac{1}{a} + frac{1}{b} + frac{1}{c} = a + b+ c$$
Prove that
$$frac{1}{(2a+b+c)^2} + frac{1}{(2a+b+c)^2} + frac{1}{(2a+b+c)^2} leqslant frac{3}{16} $$
In the solution given by the official short list solution book (Pg 16), it states that
Without loss of generality, we choose $$a +b+c = 1$$.
Thus, the problem becomes
$$ frac{1}{(1 + a)^2}
+ frac{1}{(1 + b)^2}
+frac{1}{(1 + b)^2}
leqslant frac{3}{16}(frac{1}{a} + frac{1}{b} + frac{1}{c})$$
Applying Jensen’s inequality to the function $$f(x) = frac{x}{ (1+ x)^2}$$
,which is concave for $0 ≤ x ≤ 2$
and increasing for $0 ≤ x ≤ 1$, we obtain
$$α
frac{a}{(1 + a)^2} + β
frac{b}{(1 + b)^2} + γ
frac{c}{(1 + c)^2} leqslant
(α + β + γ)
frac{A}{(1 + A)^2}$$
, where $A =frac{αa + βb + γc}{α + β + γ}.$
Choosing $α =
frac{1}{a}
, β =frac{1}{b},$ and $γ = frac{1}{c}$
, we can apply the harmonic-arithmetic-mean inequality
$$A =frac{3}{frac{1}{a} + frac{1}{b} + frac{1}{c}}
≤
frac{a + b + c
}{3}
=
frac{1}{3}
< 1$$
Finally we prove:
$$ frac{1}{(1 + a)^2}
+ frac{1}{(1 + b)^2}
+frac{1}{(1 + b)^2}
leqslant ( frac{1}{a} + frac{1}{b} + frac{1}{c}) frac{A}{(1 + A)^2} leqslant ( frac{1}{a} + frac{1}{b} + frac{1}{c})frac{A}{(1 + frac{1}{3})^2} = frac{3}{16}(frac{1}{a} + frac{1}{b} + frac{1}{c}) $$
However, Evan Chen's solution (Pg 4) differs by having $a +b+c=3$ and setting $f(x) = frac{1}{16x} - frac{1}{(x+3)^2}$ and allowing Jensen to resolve the rest.
The questions are as follows;
How do you choose before hand which $a + b+ c =$ to choose?
Why particularly does the assumption differ between the 2 solutions?
How does one know before hand the choosing of $α =
frac{1}{a}
, β =frac{1}{b},$ and $γ = frac{1}{c}$?Where and how did $A =frac{αa + βb + γc}{α + β + γ}$ come from?
Any help would be much appreciated.
inequality contest-math
inequality contest-math
edited Nov 16 at 1:48
asked Nov 15 at 19:45
299792458
236
236
I have found another solution. If you want I am ready to post it.
– Michael Rozenberg
Nov 15 at 20:03
Here is a problem for me, since $$(a+b+c)({1over a}+{1over b}+{1over c})geq 9$$ we have $a+b+cgeq 3$, so I don't understand either solution.
– greedoid
Nov 15 at 20:19
@MichaelRozenberg Can you explain please this phenomena?
– greedoid
Nov 15 at 20:23
@greedoid After homogenization we need to prove that $sumlimits_{cyc}frac{1}{(2a+b+c)^2}leqfrac{3}{16(a+b+c)}sumlimits_{cyc}frac{1}{a}$ and we can assume that even $a+b+c=17$, but we saw that the assuming was $a+b+c=1.$ In your example just $sumlimits_{cyc}frac{1}{a}geq9.$
– Michael Rozenberg
Nov 15 at 20:35
@greedoid See please the first inequality in my previous post. This inequality is equivalent to the given inequality, but it's homogeneous already.
– Michael Rozenberg
Nov 15 at 20:40
|
show 6 more comments
I have found another solution. If you want I am ready to post it.
– Michael Rozenberg
Nov 15 at 20:03
Here is a problem for me, since $$(a+b+c)({1over a}+{1over b}+{1over c})geq 9$$ we have $a+b+cgeq 3$, so I don't understand either solution.
– greedoid
Nov 15 at 20:19
@MichaelRozenberg Can you explain please this phenomena?
– greedoid
Nov 15 at 20:23
@greedoid After homogenization we need to prove that $sumlimits_{cyc}frac{1}{(2a+b+c)^2}leqfrac{3}{16(a+b+c)}sumlimits_{cyc}frac{1}{a}$ and we can assume that even $a+b+c=17$, but we saw that the assuming was $a+b+c=1.$ In your example just $sumlimits_{cyc}frac{1}{a}geq9.$
– Michael Rozenberg
Nov 15 at 20:35
@greedoid See please the first inequality in my previous post. This inequality is equivalent to the given inequality, but it's homogeneous already.
– Michael Rozenberg
Nov 15 at 20:40
I have found another solution. If you want I am ready to post it.
– Michael Rozenberg
Nov 15 at 20:03
I have found another solution. If you want I am ready to post it.
– Michael Rozenberg
Nov 15 at 20:03
Here is a problem for me, since $$(a+b+c)({1over a}+{1over b}+{1over c})geq 9$$ we have $a+b+cgeq 3$, so I don't understand either solution.
– greedoid
Nov 15 at 20:19
Here is a problem for me, since $$(a+b+c)({1over a}+{1over b}+{1over c})geq 9$$ we have $a+b+cgeq 3$, so I don't understand either solution.
– greedoid
Nov 15 at 20:19
@MichaelRozenberg Can you explain please this phenomena?
– greedoid
Nov 15 at 20:23
@MichaelRozenberg Can you explain please this phenomena?
– greedoid
Nov 15 at 20:23
@greedoid After homogenization we need to prove that $sumlimits_{cyc}frac{1}{(2a+b+c)^2}leqfrac{3}{16(a+b+c)}sumlimits_{cyc}frac{1}{a}$ and we can assume that even $a+b+c=17$, but we saw that the assuming was $a+b+c=1.$ In your example just $sumlimits_{cyc}frac{1}{a}geq9.$
– Michael Rozenberg
Nov 15 at 20:35
@greedoid After homogenization we need to prove that $sumlimits_{cyc}frac{1}{(2a+b+c)^2}leqfrac{3}{16(a+b+c)}sumlimits_{cyc}frac{1}{a}$ and we can assume that even $a+b+c=17$, but we saw that the assuming was $a+b+c=1.$ In your example just $sumlimits_{cyc}frac{1}{a}geq9.$
– Michael Rozenberg
Nov 15 at 20:35
@greedoid See please the first inequality in my previous post. This inequality is equivalent to the given inequality, but it's homogeneous already.
– Michael Rozenberg
Nov 15 at 20:40
@greedoid See please the first inequality in my previous post. This inequality is equivalent to the given inequality, but it's homogeneous already.
– Michael Rozenberg
Nov 15 at 20:40
|
show 6 more comments
2 Answers
2
active
oldest
votes
up vote
2
down vote
In a homogeneous inequality, this doesn't matter; and $ldots$
$ldots$ there is little difference between the two solutions in this respect, because if you take $a, b, c$ from Problem Shortlist with Solutions [it's on p.14 of the PDF you linked to, by the way, rather than p.16] and write $a' = 3a$, $b' = 3b$, $c' = 3c$, then $a', b', c'$ are the $a, b, c$ of Chen's solution, which homogenises the inequality in exactly the same way.
Given the idea of applying Jensen's inequality to $x/(1 + x)^2$, the choice of weights $alpha = 1/a$, $beta = 1/b$, $gamma = 1/c$ then gives you the LHS of the desired inequality (in its transformed homogeneous form).
The expression $frac{αa + βb + γc}{α + β + γ}$ will naturally appear in any application of Jensen's inequality to a function evaluated at $a, b, c$ with weights $alpha, beta, gamma$.
3. But how do you know before hand (as you were solving the problem for the first time) that the specific choice of weights were as stated? Did it had to do with the initial conditions of $frac{1}{a} + frac{1}{b} +frac{1}{c}$? And how does one derive it from here?
– 299792458
Nov 16 at 2:12
4. The same goes for $A$. How do you derive it?
– 299792458
Nov 16 at 2:14
Unaccountably, I seem to have failed to recognise the important qualification beforehand in both parts 1 and 3 of the question - thus really missing the point of your question altogether! I'm sorry. Also, I don't have any light to shed on how one knows when to apply Jensen's inequality. I don't think I'll go so far as to delete my answer, but I think it is probably best ignored!
– Calum Gilhooley
Nov 16 at 2:25
add a comment |
up vote
0
down vote
Another solution.
We need to show that:
$$frac{3}{16}-sum_{cyc}frac{1}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}left(frac{3}{16a(a+b+c)}-frac{1}{(2a+b+c)^2}right)geq0$$ or
$$sum_{cyc}frac{(3(2a+b+c)^2-16a(a+b+c))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(3b^2+3c^2-6a^2-4ab-4ac+6bc)bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(2a+3b+3c)((c-a)-(a-b))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}(a-b)left(frac{(2b+3c+3a)ca}{(2b+c+a)^2}-frac{(2a+3b+3c)bc}{(2a+b+c)^2}right)geq0,$$ which is obvious because
$$(a-b)left((2b+3c+3a)a-(2a+3b+3c)bright)geq0$$ and
$$(a-b)((2a+b+c)^2-(2b+a+c)^2)geq0.$$
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
In a homogeneous inequality, this doesn't matter; and $ldots$
$ldots$ there is little difference between the two solutions in this respect, because if you take $a, b, c$ from Problem Shortlist with Solutions [it's on p.14 of the PDF you linked to, by the way, rather than p.16] and write $a' = 3a$, $b' = 3b$, $c' = 3c$, then $a', b', c'$ are the $a, b, c$ of Chen's solution, which homogenises the inequality in exactly the same way.
Given the idea of applying Jensen's inequality to $x/(1 + x)^2$, the choice of weights $alpha = 1/a$, $beta = 1/b$, $gamma = 1/c$ then gives you the LHS of the desired inequality (in its transformed homogeneous form).
The expression $frac{αa + βb + γc}{α + β + γ}$ will naturally appear in any application of Jensen's inequality to a function evaluated at $a, b, c$ with weights $alpha, beta, gamma$.
3. But how do you know before hand (as you were solving the problem for the first time) that the specific choice of weights were as stated? Did it had to do with the initial conditions of $frac{1}{a} + frac{1}{b} +frac{1}{c}$? And how does one derive it from here?
– 299792458
Nov 16 at 2:12
4. The same goes for $A$. How do you derive it?
– 299792458
Nov 16 at 2:14
Unaccountably, I seem to have failed to recognise the important qualification beforehand in both parts 1 and 3 of the question - thus really missing the point of your question altogether! I'm sorry. Also, I don't have any light to shed on how one knows when to apply Jensen's inequality. I don't think I'll go so far as to delete my answer, but I think it is probably best ignored!
– Calum Gilhooley
Nov 16 at 2:25
add a comment |
up vote
2
down vote
In a homogeneous inequality, this doesn't matter; and $ldots$
$ldots$ there is little difference between the two solutions in this respect, because if you take $a, b, c$ from Problem Shortlist with Solutions [it's on p.14 of the PDF you linked to, by the way, rather than p.16] and write $a' = 3a$, $b' = 3b$, $c' = 3c$, then $a', b', c'$ are the $a, b, c$ of Chen's solution, which homogenises the inequality in exactly the same way.
Given the idea of applying Jensen's inequality to $x/(1 + x)^2$, the choice of weights $alpha = 1/a$, $beta = 1/b$, $gamma = 1/c$ then gives you the LHS of the desired inequality (in its transformed homogeneous form).
The expression $frac{αa + βb + γc}{α + β + γ}$ will naturally appear in any application of Jensen's inequality to a function evaluated at $a, b, c$ with weights $alpha, beta, gamma$.
3. But how do you know before hand (as you were solving the problem for the first time) that the specific choice of weights were as stated? Did it had to do with the initial conditions of $frac{1}{a} + frac{1}{b} +frac{1}{c}$? And how does one derive it from here?
– 299792458
Nov 16 at 2:12
4. The same goes for $A$. How do you derive it?
– 299792458
Nov 16 at 2:14
Unaccountably, I seem to have failed to recognise the important qualification beforehand in both parts 1 and 3 of the question - thus really missing the point of your question altogether! I'm sorry. Also, I don't have any light to shed on how one knows when to apply Jensen's inequality. I don't think I'll go so far as to delete my answer, but I think it is probably best ignored!
– Calum Gilhooley
Nov 16 at 2:25
add a comment |
up vote
2
down vote
up vote
2
down vote
In a homogeneous inequality, this doesn't matter; and $ldots$
$ldots$ there is little difference between the two solutions in this respect, because if you take $a, b, c$ from Problem Shortlist with Solutions [it's on p.14 of the PDF you linked to, by the way, rather than p.16] and write $a' = 3a$, $b' = 3b$, $c' = 3c$, then $a', b', c'$ are the $a, b, c$ of Chen's solution, which homogenises the inequality in exactly the same way.
Given the idea of applying Jensen's inequality to $x/(1 + x)^2$, the choice of weights $alpha = 1/a$, $beta = 1/b$, $gamma = 1/c$ then gives you the LHS of the desired inequality (in its transformed homogeneous form).
The expression $frac{αa + βb + γc}{α + β + γ}$ will naturally appear in any application of Jensen's inequality to a function evaluated at $a, b, c$ with weights $alpha, beta, gamma$.
In a homogeneous inequality, this doesn't matter; and $ldots$
$ldots$ there is little difference between the two solutions in this respect, because if you take $a, b, c$ from Problem Shortlist with Solutions [it's on p.14 of the PDF you linked to, by the way, rather than p.16] and write $a' = 3a$, $b' = 3b$, $c' = 3c$, then $a', b', c'$ are the $a, b, c$ of Chen's solution, which homogenises the inequality in exactly the same way.
Given the idea of applying Jensen's inequality to $x/(1 + x)^2$, the choice of weights $alpha = 1/a$, $beta = 1/b$, $gamma = 1/c$ then gives you the LHS of the desired inequality (in its transformed homogeneous form).
The expression $frac{αa + βb + γc}{α + β + γ}$ will naturally appear in any application of Jensen's inequality to a function evaluated at $a, b, c$ with weights $alpha, beta, gamma$.
answered Nov 15 at 23:50
Calum Gilhooley
4,052529
4,052529
3. But how do you know before hand (as you were solving the problem for the first time) that the specific choice of weights were as stated? Did it had to do with the initial conditions of $frac{1}{a} + frac{1}{b} +frac{1}{c}$? And how does one derive it from here?
– 299792458
Nov 16 at 2:12
4. The same goes for $A$. How do you derive it?
– 299792458
Nov 16 at 2:14
Unaccountably, I seem to have failed to recognise the important qualification beforehand in both parts 1 and 3 of the question - thus really missing the point of your question altogether! I'm sorry. Also, I don't have any light to shed on how one knows when to apply Jensen's inequality. I don't think I'll go so far as to delete my answer, but I think it is probably best ignored!
– Calum Gilhooley
Nov 16 at 2:25
add a comment |
3. But how do you know before hand (as you were solving the problem for the first time) that the specific choice of weights were as stated? Did it had to do with the initial conditions of $frac{1}{a} + frac{1}{b} +frac{1}{c}$? And how does one derive it from here?
– 299792458
Nov 16 at 2:12
4. The same goes for $A$. How do you derive it?
– 299792458
Nov 16 at 2:14
Unaccountably, I seem to have failed to recognise the important qualification beforehand in both parts 1 and 3 of the question - thus really missing the point of your question altogether! I'm sorry. Also, I don't have any light to shed on how one knows when to apply Jensen's inequality. I don't think I'll go so far as to delete my answer, but I think it is probably best ignored!
– Calum Gilhooley
Nov 16 at 2:25
3. But how do you know before hand (as you were solving the problem for the first time) that the specific choice of weights were as stated? Did it had to do with the initial conditions of $frac{1}{a} + frac{1}{b} +frac{1}{c}$? And how does one derive it from here?
– 299792458
Nov 16 at 2:12
3. But how do you know before hand (as you were solving the problem for the first time) that the specific choice of weights were as stated? Did it had to do with the initial conditions of $frac{1}{a} + frac{1}{b} +frac{1}{c}$? And how does one derive it from here?
– 299792458
Nov 16 at 2:12
4. The same goes for $A$. How do you derive it?
– 299792458
Nov 16 at 2:14
4. The same goes for $A$. How do you derive it?
– 299792458
Nov 16 at 2:14
Unaccountably, I seem to have failed to recognise the important qualification beforehand in both parts 1 and 3 of the question - thus really missing the point of your question altogether! I'm sorry. Also, I don't have any light to shed on how one knows when to apply Jensen's inequality. I don't think I'll go so far as to delete my answer, but I think it is probably best ignored!
– Calum Gilhooley
Nov 16 at 2:25
Unaccountably, I seem to have failed to recognise the important qualification beforehand in both parts 1 and 3 of the question - thus really missing the point of your question altogether! I'm sorry. Also, I don't have any light to shed on how one knows when to apply Jensen's inequality. I don't think I'll go so far as to delete my answer, but I think it is probably best ignored!
– Calum Gilhooley
Nov 16 at 2:25
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Another solution.
We need to show that:
$$frac{3}{16}-sum_{cyc}frac{1}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}left(frac{3}{16a(a+b+c)}-frac{1}{(2a+b+c)^2}right)geq0$$ or
$$sum_{cyc}frac{(3(2a+b+c)^2-16a(a+b+c))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(3b^2+3c^2-6a^2-4ab-4ac+6bc)bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(2a+3b+3c)((c-a)-(a-b))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}(a-b)left(frac{(2b+3c+3a)ca}{(2b+c+a)^2}-frac{(2a+3b+3c)bc}{(2a+b+c)^2}right)geq0,$$ which is obvious because
$$(a-b)left((2b+3c+3a)a-(2a+3b+3c)bright)geq0$$ and
$$(a-b)((2a+b+c)^2-(2b+a+c)^2)geq0.$$
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Another solution.
We need to show that:
$$frac{3}{16}-sum_{cyc}frac{1}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}left(frac{3}{16a(a+b+c)}-frac{1}{(2a+b+c)^2}right)geq0$$ or
$$sum_{cyc}frac{(3(2a+b+c)^2-16a(a+b+c))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(3b^2+3c^2-6a^2-4ab-4ac+6bc)bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(2a+3b+3c)((c-a)-(a-b))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}(a-b)left(frac{(2b+3c+3a)ca}{(2b+c+a)^2}-frac{(2a+3b+3c)bc}{(2a+b+c)^2}right)geq0,$$ which is obvious because
$$(a-b)left((2b+3c+3a)a-(2a+3b+3c)bright)geq0$$ and
$$(a-b)((2a+b+c)^2-(2b+a+c)^2)geq0.$$
add a comment |
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up vote
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down vote
Another solution.
We need to show that:
$$frac{3}{16}-sum_{cyc}frac{1}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}left(frac{3}{16a(a+b+c)}-frac{1}{(2a+b+c)^2}right)geq0$$ or
$$sum_{cyc}frac{(3(2a+b+c)^2-16a(a+b+c))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(3b^2+3c^2-6a^2-4ab-4ac+6bc)bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(2a+3b+3c)((c-a)-(a-b))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}(a-b)left(frac{(2b+3c+3a)ca}{(2b+c+a)^2}-frac{(2a+3b+3c)bc}{(2a+b+c)^2}right)geq0,$$ which is obvious because
$$(a-b)left((2b+3c+3a)a-(2a+3b+3c)bright)geq0$$ and
$$(a-b)((2a+b+c)^2-(2b+a+c)^2)geq0.$$
Another solution.
We need to show that:
$$frac{3}{16}-sum_{cyc}frac{1}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}left(frac{3}{16a(a+b+c)}-frac{1}{(2a+b+c)^2}right)geq0$$ or
$$sum_{cyc}frac{(3(2a+b+c)^2-16a(a+b+c))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(3b^2+3c^2-6a^2-4ab-4ac+6bc)bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}frac{(2a+3b+3c)((c-a)-(a-b))bc}{(2a+b+c)^2}geq0$$ or
$$sum_{cyc}(a-b)left(frac{(2b+3c+3a)ca}{(2b+c+a)^2}-frac{(2a+3b+3c)bc}{(2a+b+c)^2}right)geq0,$$ which is obvious because
$$(a-b)left((2b+3c+3a)a-(2a+3b+3c)bright)geq0$$ and
$$(a-b)((2a+b+c)^2-(2b+a+c)^2)geq0.$$
answered Nov 16 at 5:19
Michael Rozenberg
94.3k1588183
94.3k1588183
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I have found another solution. If you want I am ready to post it.
– Michael Rozenberg
Nov 15 at 20:03
Here is a problem for me, since $$(a+b+c)({1over a}+{1over b}+{1over c})geq 9$$ we have $a+b+cgeq 3$, so I don't understand either solution.
– greedoid
Nov 15 at 20:19
@MichaelRozenberg Can you explain please this phenomena?
– greedoid
Nov 15 at 20:23
@greedoid After homogenization we need to prove that $sumlimits_{cyc}frac{1}{(2a+b+c)^2}leqfrac{3}{16(a+b+c)}sumlimits_{cyc}frac{1}{a}$ and we can assume that even $a+b+c=17$, but we saw that the assuming was $a+b+c=1.$ In your example just $sumlimits_{cyc}frac{1}{a}geq9.$
– Michael Rozenberg
Nov 15 at 20:35
@greedoid See please the first inequality in my previous post. This inequality is equivalent to the given inequality, but it's homogeneous already.
– Michael Rozenberg
Nov 15 at 20:40