Difference between revisions of "2004 USAMO Problems/Problem 5"

Revision as of 23:11, 1 July 2008

Problem 5

(Titu Andreescu) Let $\displaystyle a$ , $\displaystyle b$ , and $\displaystyle c$ be positive real numbers. Prove that

$(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3$ .

Solutions

We first note that for positive $x$ , $x^5 + 1 \ge x^3 + x^2$ . We may prove this in the following ways:

Since $x^2$ and $x^3$ must be both lesser than, both equal to, or both greater than 1, by the rearrangement inequality, $x^2 \cdot x^3 + 1 \cdot 1 \ge x^2 \cdot 1 + 1 \cdot x^3$ .

Since $x^2 - 1$ and $x^3 - 1$ have the same sign, $0 \le (x^2 - 1)(x^3 - 1) = x^5 - x^3 - x^2 + 1$ , with equality when $x = 1$ .

By weighted AM-GM, $\frac{2}{5}x^5 + \frac{3}{5} \ge x^2$ and $\frac{3}{5}x^5 + \frac{2}{5} \ge x^3$ . Adding these gives the desired inequality. Equivalently, the desired inequality is a case of Muirhead's Inequality.

It thus becomes sufficient to prove that

$(a^3 + 2)(b^3 + 2)(c^3 + 2) \ge (a+b+c)^3$ .

This follows from Hölder's Inequality

$\begin{matrix}(m_{1,1} + m_{1,2} + m_{1,3})(m_{2,1} + m_{2,2} + m_{2,3})(m_{3,1} + m_{3,2} + m_{3,3}) \ge \\ \qquad \qquad \qquad\left[ (m_{1,1}m_{2,1}m_{3,1})^{1/3} + (m_{2,1}m_{2,2}m_{2,3})^{1/3} + (m_{3,1}m_{3,2}m_{3,3})^{1/3} \right] ^3 \end{matrix}$

when we take $m_{1,1} = a^3$ , $m_{2,2} = b^3$ , $m_{3,3} = c^3$ , and $m_{x,y} = 1$ when $x \neq y$ . We have equality if and only if $a = b = c = 1$ .

It is also possible to solve this inequality by expanding terms and applying brute force, either before or after proving that $x^5 - x^2 + 3 \ge x^3 + 2$ .

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

Resources

@@ Line 26: / Line 26: @@
 </center>
-We present two proofs of this inequality.
+This follows from [[Hölder's Inequality]]
-First, [[Hölder's Inequality]] gives us
 <center>
 <math>
 \begin{matrix}(m_{1,1} + m_{1,2} + m_{1,3})(m_{2,1} + m_{2,2} + m_{2,3})(m_{3,1} + m_{3,2} + m_{3,3}) \ge \\ \qquad \qquad \qquad\left[ (m_{1,1}m_{2,1}m_{3,1})^{1/3} + (m_{2,1}m_{2,2}m_{2,3})^{1/3} + (m_{3,1}m_{3,2}m_{3,3})^{1/3} \right] ^3 \end{matrix}
-</math>.
+</math>
 </center>
-Setting <math>a = m_{1,1} </math>, <math>b = m_{2,2} </math>, <math>c = m_{3,3} </math>, and <math>m_{x,y} = 1 </math> when <math> x \neq y </math> gives us the desired inequality, with equality when <math>x = y = z = 1 </math>.
+when we take <math>m_{1,1} = a^3</math>, <math>m_{2,2} = b^3</math>, <math>m_{3,3} = c^3</math>, and <math>m_{x,y} = 1 </math> when <math> x \neq y </math>. We have equality if and only if <math>a = b = c = 1 </math>.
-Second, we may apply the [[Cauchy-Schwarz Inequality]] twice to obtain
+''It is also possible to solve this inequality by expanding terms and applying brute force, either before or after proving that <math>x^5 - x^2 + 3 \ge x^3 + 2 </math>.''
-<center>
-<math>
-\begin{matrix}
-\left[(a^2 + 1 + 1)(1 + b^2 + 1)\right] \left[ (1 + 1 + c^2)(a + b + c) \right] & \ge & (a + b + 1)^2( a + b + c^2)^2 \\
-& \ge & (a + b + c)^4 \qquad \qquad \quad \; \;
-\end{matrix}
-</math>,
-</center>
-as desired.
-''Unfortunately, it is also possible to solve this inequality by expanding terms and applying brute force, either before or after proving that <math>x^5 - x^2 + 3 \ge x^3 + 2 </math>.''

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Difference between revisions of "2004 USAMO Problems/Problem 5"

Revision as of 23:11, 1 July 2008

Problem 5

Solutions

Resources