Inequalities (Thailand Math POSN 1st elimination round 2014)

Only use any the following.

  • Basic inequalities
  • AM-GM-HM
  • Holder
  • Cauchy Schwarz (Titu's lemma is acceptable)
  • Weighted AM-GM

1.) Let a,b,ca,b,c be positive real numbers such that abc=1abc = 1. Prove that

4(a+2)2+4b2+4(b+2)2+4c2+4(c+2)2+4a21\frac{4}{(a+2)^{2}+4b^{2}} + \frac{4}{(b+2)^{2}+4c^{2}} + \frac{4}{(c+2)^{2}+4a^{2}} \leq 1

2.) Find all positive real solutions a,b,ca,b,c such that

{10a3+b3=7ab+20bc11b3+20c3=7bc+30ca2034c3+44a3=51ca+20ab50 \begin{cases} 10a^{3}+b^{3} = 7ab + 20bc - 11 \\ b^{3}+20c^{3} = 7bc + 30ca - 20 \\ 34c^{3}+44a^{3} = 51ca + 20ab - 50 \\ \end{cases}

3.) Let a,b,ca,b,c be positive real numbers such that abc=1abc = 1. Prove that

a3b+b3c+c3aa2/5b3/5+b2/5c3/5+c2/5a3/5a^{3}b+b^{3}c+c^{3}a \geq a^{2/5}b^{3/5} + b^{2/5}c^{3/5} + c^{2/5}a^{3/5}

4.) Let a,b,ca,b,c be positive real numbers such that a+b+c+abc=4a+b+c+abc = 4. Prove that

(ab+c+bc+a+ca+b)2(ab+bc+ca)12(4abc)3\left(\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b}}\right)^{2}(ab+bc+ca) \geq \frac{1}{2}(4-abc)^{3}

5.) Let a,b,ca,b,c be positive real numbers. Prove that

a9bc+b9ca+c9ab+3abca4+b4+c4+3\frac{a^{9}}{bc}+\frac{b^{9}}{ca}+\frac{c^{9}}{ab} + \frac{3}{abc} \geq a^{4}+b^{4}+c^{4}+3

Check out all my notes and stuffs for more problems!

Thailand Math POSN 2013

Thailand Math POSN 2014

Note by Samuraiwarm Tsunayoshi
4 years, 12 months ago

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1 vote

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I like the kind of questions where a suitably application of AM-GM gives the resultant inequality, since that relies on your ability to spot / identify patterns.

For example, here is how to do 3:

Apply AM-GM to 2a3b+2b3c+c3a 2 a^3 b + 2 b^3 c + c^3 a .

Calvin Lin Staff - 4 years, 11 months ago

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Multiply abcabc on the RHS. Then Muirhead.

Muirheaders gonna Muirhead Muirhead Muirhead

Daniel Liu - 4 years, 11 months ago

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Right. Muirhead is just a fancy term for bunching AM-GM.

Calvin Lin Staff - 4 years, 11 months ago

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@Calvin Lin In addition, graders hate it because it involves no ingenuity at all.

Daniel Liu - 4 years, 11 months ago

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Sorry I miss typed no.4 here. It is a+b+c+abc=4a+b+c+abc = 4 instead of 11.

Samuraiwarm Tsunayoshi - 4 years, 11 months ago

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Is it necessary to use just the ineqs u stated?

Dinesh Chavan - 4 years, 11 months ago

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Yup.

Samuraiwarm Tsunayoshi - 4 years, 11 months ago

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