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prove this inequality $sumfracabge sum a^2$ [closed]
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Let $a,b,c>0$ such $a+b+c=3$, show that
$$dfracab+dfracbc+dfraccage a^2+b^2+c^2$$
I have show this not stronger inequality
$$sumdfracabge sum a$$
inequality
asked Jul 18 at 10:40
wightahtl
12412
closed as off-topic by Alex Francisco, Isaac Browne, amWhy, Adrian Keister, user223391 Jul 19 at 14:06
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Isaac Browne, amWhy, Adrian Keister, Community
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up vote
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Let $a,b,c>0$ such $a+b+c=3$, show that
$$dfracab+dfracbc+dfraccage a^2+b^2+c^2$$
I have show this not stronger inequality
$$sumdfracabge sum a$$
inequality
asked Jul 18 at 10:40
wightahtl
12412
closed as off-topic by Alex Francisco, Isaac Browne, amWhy, Adrian Keister, user223391 Jul 19 at 14:06
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Isaac Browne, amWhy, Adrian Keister, Community
1
What is the range of your sums? What have you done so far? Can you explain more?
– Shervin Sorouri
Jul 18 at 10:46
Use the $uvw$ technique.
– Michael Rozenberg
Jul 18 at 21:11
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $a,b,c>0$ such $a+b+c=3$, show that
$$dfracab+dfracbc+dfraccage a^2+b^2+c^2$$
I have show this not stronger inequality
$$sumdfracabge sum a$$
inequality
asked Jul 18 at 10:40
wightahtl
12412
Let $a,b,c>0$ such $a+b+c=3$, show that
$$dfracab+dfracbc+dfraccage a^2+b^2+c^2$$
I have show this not stronger inequality
$$sumdfracabge sum a$$
inequality
asked Jul 18 at 10:40
wightahtl
12412
asked Jul 18 at 10:40
wightahtl
12412
asked Jul 18 at 10:40
wightahtl
12412
asked Jul 18 at 10:40
wightahtl
12412
12412
closed as off-topic by Alex Francisco, Isaac Browne, amWhy, Adrian Keister, user223391 Jul 19 at 14:06
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Isaac Browne, amWhy, Adrian Keister, Community
closed as off-topic by Alex Francisco, Isaac Browne, amWhy, Adrian Keister, user223391 Jul 19 at 14:06
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Isaac Browne, amWhy, Adrian Keister, Community
1
What is the range of your sums? What have you done so far? Can you explain more?
– Shervin Sorouri
Jul 18 at 10:46
Use the $uvw$ technique.
– Michael Rozenberg
Jul 18 at 21:11
add a comment |Â
1
What is the range of your sums? What have you done so far? Can you explain more?
– Shervin Sorouri
Jul 18 at 10:46
Use the $uvw$ technique.
– Michael Rozenberg
Jul 18 at 21:11
1
1
What is the range of your sums? What have you done so far? Can you explain more?
– Shervin Sorouri
Jul 18 at 10:46
What is the range of your sums? What have you done so far? Can you explain more?
– Shervin Sorouri
Jul 18 at 10:46
Use the $uvw$ technique.
– Michael Rozenberg
Jul 18 at 21:11
Use the $uvw$ technique.
– Michael Rozenberg
Jul 18 at 21:11
add a comment |Â
1 Answer
1
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Let $sum $ denote cyclic sum, then observe:
$$sum a(a-b)^2(b-2c)^2 geqslant 0$$
$$implies sum ab^4 + sum a^3b^2 + 2sum a^2b^3+4abcsum ab -8abcsum a^2 geqslant 0 $$
$$implies 2left( sum a right)^2sum ab^2 + abcleft(sum aright)^2 geqslant 21abcsum a^2$$
$$implies 6 sum fracab + 3 geqslant 7sum a^2$$
Add the obvious $sum a^2 geqslant 3$ to the above to conclude. Equality is when $a=b=c=1$.
answered Jul 19 at 9:41
Macavity
34.4k52351
Yes, it's a very known solution, but I think it's impossible to find this solution during a competition. By the way, the $uvw$'s technique gets a smooth proof.
– Michael Rozenberg
Jul 19 at 15:18
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Let $sum $ denote cyclic sum, then observe:
$$sum a(a-b)^2(b-2c)^2 geqslant 0$$
$$implies sum ab^4 + sum a^3b^2 + 2sum a^2b^3+4abcsum ab -8abcsum a^2 geqslant 0 $$
$$implies 2left( sum a right)^2sum ab^2 + abcleft(sum aright)^2 geqslant 21abcsum a^2$$
$$implies 6 sum fracab + 3 geqslant 7sum a^2$$
Add the obvious $sum a^2 geqslant 3$ to the above to conclude. Equality is when $a=b=c=1$.
answered Jul 19 at 9:41
Macavity
34.4k52351
Yes, it's a very known solution, but I think it's impossible to find this solution during a competition. By the way, the $uvw$'s technique gets a smooth proof.
– Michael Rozenberg
Jul 19 at 15:18
add a comment |Â
up vote
1
down vote
Let $sum $ denote cyclic sum, then observe:
$$sum a(a-b)^2(b-2c)^2 geqslant 0$$
$$implies sum ab^4 + sum a^3b^2 + 2sum a^2b^3+4abcsum ab -8abcsum a^2 geqslant 0 $$
$$implies 2left( sum a right)^2sum ab^2 + abcleft(sum aright)^2 geqslant 21abcsum a^2$$
$$implies 6 sum fracab + 3 geqslant 7sum a^2$$
Add the obvious $sum a^2 geqslant 3$ to the above to conclude. Equality is when $a=b=c=1$.
answered Jul 19 at 9:41
Macavity
34.4k52351
Yes, it's a very known solution, but I think it's impossible to find this solution during a competition. By the way, the $uvw$'s technique gets a smooth proof.
– Michael Rozenberg
Jul 19 at 15:18
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Let $sum $ denote cyclic sum, then observe:
$$sum a(a-b)^2(b-2c)^2 geqslant 0$$
$$implies sum ab^4 + sum a^3b^2 + 2sum a^2b^3+4abcsum ab -8abcsum a^2 geqslant 0 $$
$$implies 2left( sum a right)^2sum ab^2 + abcleft(sum aright)^2 geqslant 21abcsum a^2$$
$$implies 6 sum fracab + 3 geqslant 7sum a^2$$
Add the obvious $sum a^2 geqslant 3$ to the above to conclude. Equality is when $a=b=c=1$.
answered Jul 19 at 9:41
Macavity
34.4k52351
Let $sum $ denote cyclic sum, then observe:
$$sum a(a-b)^2(b-2c)^2 geqslant 0$$
$$implies sum ab^4 + sum a^3b^2 + 2sum a^2b^3+4abcsum ab -8abcsum a^2 geqslant 0 $$
$$implies 2left( sum a right)^2sum ab^2 + abcleft(sum aright)^2 geqslant 21abcsum a^2$$
$$implies 6 sum fracab + 3 geqslant 7sum a^2$$
Add the obvious $sum a^2 geqslant 3$ to the above to conclude. Equality is when $a=b=c=1$.
answered Jul 19 at 9:41
Macavity
34.4k52351
answered Jul 19 at 9:41
Macavity
34.4k52351
answered Jul 19 at 9:41
Macavity
34.4k52351
answered Jul 19 at 9:41
Macavity
34.4k52351
34.4k52351
Yes, it's a very known solution, but I think it's impossible to find this solution during a competition. By the way, the $uvw$'s technique gets a smooth proof.
– Michael Rozenberg
Jul 19 at 15:18
add a comment |Â
Yes, it's a very known solution, but I think it's impossible to find this solution during a competition. By the way, the $uvw$'s technique gets a smooth proof.
– Michael Rozenberg
Jul 19 at 15:18
Yes, it's a very known solution, but I think it's impossible to find this solution during a competition. By the way, the $uvw$'s technique gets a smooth proof.
– Michael Rozenberg
Jul 19 at 15:18
Yes, it's a very known solution, but I think it's impossible to find this solution during a competition. By the way, the $uvw$'s technique gets a smooth proof.
– Michael Rozenberg
Jul 19 at 15:18
add a comment |Â
1
What is the range of your sums? What have you done so far? Can you explain more?
– Shervin Sorouri
Jul 18 at 10:46
Use the $uvw$ technique.
– Michael Rozenberg
Jul 18 at 21:11