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.

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1977 AHSME Problems/Problem 29

Problem 29

Solution (Official MAA)