Difference between revisions of "2001 IMO Shortlist Problems/A6"

Latest revision as of 23:12, 27 March 2021

Problem

Prove that for all positive real numbers $a,b,c$ ,

$\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}} \geq 1.$

Generalization

The leader of the Bulgarian team had come up with the following generalization to the inequality:

$\frac {a}{\sqrt {a^2 + kbc}} + \frac {b}{\sqrt {b^2 + kca}} + \frac {c}{\sqrt {c^2 + kab}} \geq \frac{3}{\sqrt{1+k}}.$

Solution

We will use the Jenson's inequality.

Now, normalize the inequality by assuming $a+b+c=1$

Consider the function $f(x)=\frac{1}{\sqrt{x}}$ . Note that this function is convex and monotonically decreasing which implies that if $a > b$ , then $f(a) < f(b)$ .

Thus, we have

$\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}} = af(a^2+8bc)+bf(b^2+8ca)+cf(c^2+8ab) \geq f(a^3+b^3+c^3+24abc)$

Thus, we only need to show that $a^3+b^3+c^3+24abc \leq 1$ i.e.

$a^3+b^3+c^3+24abc \leq (a+b+c)^3=a^3+b^3+c^3+3(a+b+c)(ab+bc+ca)-3abc$

$i.e. (a+b+c)(ab+bc+ca) \geq 9abc$

Which is true since

$(a+b+c)(ab+bc+ca) \geq (3\sqrt[3]{abc})(3\sqrt[3]{a^{2}b^{2}c^{2}}) = 9abc$ The last part follows by the AM-GM inequality.

Equality holds if $a=b=c$

Alternative Solution

By Carlson's Inequality, we can know that $\Bigg(\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}}\Bigg)\Bigg(\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}}\Bigg)\Big((a^3+8abc)+(b^3+8abc)+(c^3+8abc)\Big) \ge (a+b+c)^3$

Then, $\Bigg(\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}}\Bigg)^2 \ge \frac{(a+b+c)^3}{a^3+b^3+c^3+24abc}$

On the other hand, $3a^2b+3b^2c+3c^2a \ge 9abc$ and $3ab^2+3bc^2+3ca^2 \ge 9abc$

Then, $(a+b+c)^3 = a^3+b^3+c^3+3(a^2b+ab^2+a^2c+ac^2+b^2c+bc^2)+6abc \ge a^3+b^3+c^3+24abc$

Therefore, $\Bigg(\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}}\Bigg)^2 \ge \frac{(a+b+c)^3}{a^3+b^3+c^3+24abc} \ge 1$

Thus, $\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}} \ge 1$

-- Haozhe Yang

Resources

@@ Line 21: / Line 21: @@
 Thus, we only need to show that <math>a^3+b^3+c^3+24abc \leq 1</math> i.e.
-<cmath>a^3+b^3+c^3+24abc \leq (a+b+c)^3=a^3+b^3+c^3+3(ab+bc+ca)-3abc</cmath>
+<cmath>a^3+b^3+c^3+24abc \leq (a+b+c)^3=a^3+b^3+c^3+3(a+b+c)(ab+bc+ca)-3abc</cmath>
-<cmath>i.e.   (ab+bc+ca) \geq 9abc</cmath>
+<cmath>i.e.   (a+b+c)(ab+bc+ca) \geq 9abc</cmath>
 Which is true since
-<cmath>(ab+bc+ca)=(a+b+c)(ab+bc+ca)=(a+b)(b+c)(c+a)+abc \geq 2\sqrt{ab}.2\sqrt{bc}.2\sqrt{ca}+abc=9abc</cmath>
+<cmath>(a+b+c)(ab+bc+ca) \geq (3\sqrt[3]{abc})(3\sqrt[3]{a^{2}b^{2}c^{2}}) = 9abc</cmath>
 The last part follows by the AM-GM inequality.
 Equality holds if <math>a=b=c</math>
+== Alternative Solution ==
+By Carlson's Inequality, we can know that <cmath>\Bigg(\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}}\Bigg)\Bigg(\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}}\Bigg)\Big((a^3+8abc)+(b^3+8abc)+(c^3+8abc)\Big) \ge (a+b+c)^3</cmath>
+Then, <cmath>\Bigg(\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}}\Bigg)^2 \ge \frac{(a+b+c)^3}{a^3+b^3+c^3+24abc}</cmath>
+On the other hand, <cmath>3a^2b+3b^2c+3c^2a \ge 9abc</cmath> and <cmath>3ab^2+3bc^2+3ca^2 \ge 9abc</cmath>
+Then, <cmath>(a+b+c)^3 = a^3+b^3+c^3+3(a^2b+ab^2+a^2c+ac^2+b^2c+bc^2)+6abc \ge a^3+b^3+c^3+24abc</cmath>
+Therefore, <cmath>\Bigg(\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}}\Bigg)^2 \ge \frac{(a+b+c)^3}{a^3+b^3+c^3+24abc} \ge 1</cmath>
+Thus, <cmath>\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}} \ge 1</cmath>
+-- Haozhe Yang
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