NMO Selection Tests for the Junior Balkan Mathematical Olympiad

If $x$ is a positive integer, then either $x^{12} \equiv 0 \pmod{13}$ or $x^{12} \equiv 1 \pmod{13}$, hence $x^6$ is congruent with $-1$, $0$ or $1$ modulo $13$. Therefore, if $t$ is congruent with $6 \pmod{13}$, then $\pm x^6 + t$ is congruent with $5$, $6$ or $7 \pmod{13}$, while $\pm x^6 + t + 1$ is congruent with $6$, $7$ or $8 \pmod{13}$.

On the other hand, perfect squares are congruent with $0$, $1$, $4$, $9$, $3$, $12$ or $10 \pmod{13}$. Therefore a perfect square can not be equal to a number of the form $\pm x^6 + t$ or $\pm x^6 + t + 1$ if $t \equiv 6 \pmod{13}$. In conclusion, all the numbers that are congruent with $6$ modulo $13$ have the required property.