1977 AMC 12 Problem 29

Question

Find the smallest integer

n

such that

\begin{aligned} (x^2+y^2+z^2)^2 &\le{}\\[-4pt] &n(x^4+y^4+z^4) \end{aligned}

for all real numbers

x,

y

and

z.

2

3

4

6

There is no such integer

n.

Apply Cauchy-Schwarz to the three numbers

x^2,

y^2,

z^2

Use equal nonzero values of

x^2,

y^2,

z^2

to test sharpness

By Cauchy-Schwarz,

\begin{aligned} (x^2+y^2+z^2)^2&\le3Q,\\ Q&=x^4+y^4+z^4. \end{aligned}

Equality occurs when

x^2=y^2=z^2\ne0,

so no smaller value can work. Thus the smallest integer is

3.

Therefore, the correct answer is B.

Problem 29 in other years