With the digits $1, 2, 3,. . . . . . , 9$ three-digit numbers are written such that the sum of the three digits is $17$. How many numbers can be written?

If you multiply the number of faces that a pyramid has with the number of edges of the pyramid, you get $5.100$. Determine the number of faces of the pyramid.

The complete list of the three-digit palindrome numbers is written in ascending order: $$101, 111, 121, 131,... , 979, 989, 999.$$Then eight consecutive palindrome numbers are eliminated and the numbers that remain in the list are added, obtaining $46.150$. Determine the eight erased palindrome numbers .

In the expression $t=\frac{8a+ 1}{b}$ where $a, b, t$ are positive integers, where $b <7$. Determine the values of $a$ and$ b$ that allow to obtain $t$ under the established conditions.

Given a chord $PQ$ of a circle and $M$ the midpoint of the chord, let $AB$ and $CD$ be two chords that pass through $M$. $AC$ and $BD$ are drawn until $PQ$ is intersected at points $X$ and $Y$ respectively. Show that $X$ and $Y$ are equidistant from $M$.