There are $12$ dentists in a clinic near a school. The students of the $5$th year, who are $29$, attend the clinic. Each dentist serves at least $2$ students. Determine the greater number of students that can attend to a single dentist .

In the rectangular parallelepiped in the figure, the lengths of the segments $EH$, $HG$, and $EG$ are consecutive integers. The height of the parallelepiped is $12$. Find the volume of the parallelepiped.

With three different digits, six-digit numbers are written, multiples of $3$. One of the the digits are in the unit's place, another in the hundred's place, and the third in the remaining places. If we take out two units from the hundred's digit and add these to the unit's digit, the number is left with all the same digits. Find the numbers.

Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes.

In a trapezoid $ABCD$, the side $DA$ is perpendicular to the bases $AB$ and $CD$. Also $AB=45$, $CD =20$, $BC =65$. Let $P$ be a point on the side $BC$ such that $BP=45$ and let $M$ be the midpoint of $DA$. Calculate the length of $PM$ .