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

We see squares and one number. And we see an inequality. This calls for Cauchy's inequality. EEEEWWW.

Anyways, look at which side is which. The squared side is smaller-- so that's good. It's in the right format.

Cauchy's states that $(a_1b_1+a_2b_2+a_3b_3+......)^2 \le (a_1^2+a_2^2+a_3^2+....)(b_1^2+b_2^2+b_3^2+.....)$

Therefore, we see that, if we equate $a_1 = x^2, a_2 = y^2, a_3 = z^2$ we get the equality right away. What's the final step? Figuring out this n. Now, note that the equation is basically complete; all we need is for $b_1+b_2+b_3 = n$ . So each of them is just 1, and $n = 3$ -- answer choice $\boxed{B}$ !

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1977 AHSME Problems/Problem 29

Problem 29

Solution