1977 AHSME Problems/Problem 29

Problem 29

Find the smallest integer $n$ such that $(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)$ for all real numbers $x,y$, and $z$.

$\textbf{(A) }2\qquad \textbf{(B) }3\qquad \textbf{(C) }4\qquad \textbf{(D) }6\qquad  \textbf{(E) }\text{There is no such integer n}$

Solution

Solution (Official MAA)

Let $a = x^2$, $b = y^2$, $c = z^2$. Then

\[0 \leq (a-b)^2 + (b-c)^2 + (c-a)^2\] \[\dfrac{ab+bc+ca}{a^2 + b^2 + c^2} \leq 1;\] \[\dfrac{a^2 + b^2 + c^2 + 2(ab+bc+ca)}{a^2 + b^2 + c^2} \leq 3;\] \[(a+b+c)^2 \leq 3(a^2 + b^2 + c^2).\]

Therefore $n\leq 3$. Choosing $a = b = c > 0$ shows $n$ is not less than three.