Let $a > 0$. If the system $$\begin{cases} a^x + a^y + a^z = 14 - a \\ x + y + z = 1 \end{cases}$$has a solution in real numbers, prove that $a \le 8$.
Let us utilize AM-GM Inequality here (since $a > 0$). Based on the above pair of equations, we deduce:
$(a^x + a^y + a^z)/3 \ge (a^x * a^y * a^z)^{1/3}$;
or $14 - a \ge 3[a^{x + y + z}]^{1/3}$;
or $14 - a \ge 3a^{1/3}$;
or $0 \ge a + 3a^{1/3} - 14$;
or $0 \ge [a ^{1/3} - 2][a^{2/3} + 2a^{1/3} + 7]$ (NOTE: the RHS factor is strictly positive for all $a > 0$.);
or $a^{1/3} \le 2$;
or $a \le 8$.
QED