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Revision as of 16:59, 20 August 2008

Problem 5

(Titu Andreescu) Let $a$, $b$, and $c$ 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 $x^2 - 1$ 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

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

We present two proofs of this inequality:

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

We get the desired inequality by taking $m_{1,1} = a^3$, $m_{2,2} = b^3$, $m_{3,3} = c^3$, and $m_{x,y} = 1$ when $x \neq y$. We have equality if and only if $a = b = c = 1$.

  • Take $x = \sqrt{a}$, $y = \sqrt{b}$, and $z = \sqrt{c}$. Then some two of $x$, $y$, and $z$ are both at least $1$ or both at most $1$. Without loss of generality, say these are $x$ and $y$. Then the sequences $(x, 1, 1)$ and $(1, 1, y)$ are oppositely sorted, yielding

$(x^6 + 1 + 1)(1 + 1 + y^6) \ge 3(x^6 + 1 + y^6)$

by Chebyshev's Inequality. By the Cauchy-Schwarz Inequality we have

$(x^6 + 1 + y^6)(1 + z^6 + 1) \ge (x^3 + y^3 + z^3)^2.$

Applying Chebyshev's and the Cauchy-Schwarz Inequalities each once more, we get

$3(x^3 + y^3 + z^3) \ge (x^2 + y^2 + z^2)(x+y+z),$

and

$(x^3 + y^3 + z^3)(x+y+z) \ge (x^2 + y^2 + z^2)^2.$

Multiplying the above four inequalities together yields

$(x^6 + 2)(y^6 + 2)(z^6 + 2) \ge (x^2 + y^2 + z^2)^3,$

as desired, with equality if and only if $x = y = z = 1$.

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

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