Track / Stage 4 / 150 of 340 #410 of 1964

Problem 410

AMC 12 late, AIME early

Algebra Difficulty 4.8 Multiple choice

6. The number of elements in the set $\left\{(x, y, z) \left\lvert\, \log _{\frac{1}{4}}\left(x^{4}+y^{4}+z^{4}+1\right)\right.\right.$ $\left.\geqslant \log _{\frac{1}{4}} x+\log _{\frac{1}{4}} y+\log _{\frac{1}{4}} z-1\right\}$ is:

Pick one

Official solution

6. (B).

From the known inequality, we get $x^{4}+y^{4}+z^{4}+1 \leqslant 4 x y z$ and $x, y, z$ are all positive numbers. Also, from the important inequality, we have $x^{4}+y^{4}+z^{4}+1 \geqslant 4 x y z$ (the equality holds if and only if $x=y=z=1$ ).
$\therefore x^{4}+y^{4}+z^{4}+1=4 x y z \text {. }$

Thus, $x=y=z=1$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.