Solve the inequality

$$
\frac{1}{x}+\frac{1}{x^{3}}+\frac{1}{x^{5}}+\frac{1}{x^{7}}+\frac{1}{x^{9}}+\frac{1}{x^{11}}+\frac{1}{x^{13}} \leq \frac{7}{x^{7}}
$$
If $a$ is a positive number, then according to the inequality between the arithmetic and geometric means,

$$
\frac{1}{a}+\frac{1}{a^{3}}+\cdots+\frac{1}{a^{13}} \geq 7 \sqrt[7]{\frac{1}{a^{1+3+\cdots+13}}}=\frac{7}{a^{7}}
$$

and equality holds precisely when $\frac{1}{a}=\frac{1}{a^{3}}=\cdots=\frac{1}{a^{13}}$, that is, when $a=1$. For negative $a$, the direction of the inequality is reversed. Therefore, the inequality of the problem is satisfied by 1 and the negative numbers. Lajos Deli (Hajdúszoboszló, Hőgyes E. Gymn., 9th grade)