Interesting 3 var inequalities

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Interesting 3 var inequalities

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imnotgoodatmathsorry

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imnotgoodatmathsorry

#1 h May 17, 2025, 4:45 PM

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Problem 1. Let $x,y,z >1$ . Prove that:
$\frac{108(x^6+y^6)(y^6+z^6)(z^6+x^6)}{x^9y^9z^9} - (xy+yz+zx)^6 \le 135$
Problem 2. Let $a,b,c >1$ . Prove that:
$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$

This post has been edited 2 times. Last edited by imnotgoodatmathsorry, May 18, 2025, 11:24 PM

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#2 h May 18, 2025, 3:50 PM

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imnotgoodatmathsorry wrote:

Problem 1. Let $x,y,z >0$ . Prove that:
$\frac{108(x^6+y^6)(y^6+z^6)(z^6+x^6)}{x^9y^9z^9} - (xy+yz+zx)^6 \le 135$
Problem 2. Let $a,b,c >0$ . Prove that:
$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$

Looks bashy,but somehow there is an elegant solution for one of this.
Interesting-
(Side note:The bar for elegant has been set very low for me,so I might consider literally everything elegant-)

This post has been edited 1 time. Last edited by IceyCold, May 18, 2025, 3:52 PM

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