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Home » Olympiad Section » Inequalities » Inequalities Solved Problems
 
may be old and easy
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minhkhoa
Poincare Conjecture
Poincare Conjecture


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#1
may be old and easy
Let a,b,c be positive real numbers such that ab+bc+ca=3
Prove that
a^2+b^2+c^2+abc\ge a+b+c+1
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LNMK

PostPosted: Mon Jun 13, 2005 8:21 pm  Back to top 
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Soarer
Navier-Stokes Equations
Navier-Stokes Equations

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#2
a^2+b^2+c^2+2abc+1 \ge 2(ab+bc+ca)
a^2+b^2+c^2+3 \ge 2(a+b+c)
sum them up and done

PostPosted: Mon Jun 13, 2005 11:53 pm  Back to top 
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minhkhoa
Poincare Conjecture
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#3
siuhochung wrote:
a^2+b^2+c^2+2abc+1 \ge 2(ab+bc+ca)

Can I know why this is true
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PostPosted: Wed Jun 15, 2005 2:43 am  Back to top 
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Soarer
Navier-Stokes Equations
Navier-Stokes Equations

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#4
minhkhoa wrote:
siuhochung wrote:
a^2+b^2+c^2+2abc+1 \ge 2(ab+bc+ca)

Can I know why this is true

This is posted before, but anyway,
a^6+b^6+c^6+2a^3b^3c^3+1 \ge a^6+b^6+c^6+3a^2b^2c^2 \ge \sum_{sym} a^4b^2 \ge \sum_{sym} a^3b^3

PostPosted: Wed Jun 15, 2005 2:46 am  Back to top 
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Ji Chen
Yang-Mills Theory
Yang-Mills Theory

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#5
minhkhoa wrote:
Let a,b,c be positive real numbers such that bc + ca + ab = 3. Prove that

a^2 + b^2 + c^2 + abc\ge a + b + c + 1.
a^2 + b^2 + c^2 + abc - (a + b + c + 1) - (bc + ca + ab - 3)

= abc + a^2 + b^2 + c^2 - bc - ca - ab - a - b - c + 2

= (b - c)^2 + (a - 1)^2 + (a + 1)(b - 1)(c - 1)\geq 0,

which is clearly true for (b - 1)(c - 1) = \max\{(b - 1)(c - 1),(c - 1)(a - 1),(a - 1)(b - 1)\}.

See also : http://www.artofproblemsolving.com/Forum/viewtopic.php?t=82866

PostPosted: Sat Oct 24, 2009 12:27 am  Back to top 
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fesha3t
P versus NP
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#6
may be old
Let a, b, c>0 such that abc=8. Prove that:
\sum \frac{a^2}{\sqrt{(a^3+1)(b^3+1)}}\ge \frac{4}{3}

PostPosted: Sat Nov 14, 2009 7:06 am  Back to top 
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