Elías and Juanca solve the same problem by posing a quadratic equation. Elijah is wrong when writing the independent term and gets as results of the problem $-1$ and $-3$. Juanca is wrong only when writing the coefficient of the first degree term and gets as results of the problem $16$ and $-2$. What are the correct results of the problem?

Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small. Nair build triangular figures by following these rules: $\bullet$ Figure $1$ is made up of $4$ small pieces, Figure $2$ is made up of $2$ large pieces and $8$ small, Figure $3$ by $6$ large and $12$ small, and so on. $\bullet$ The central column must be made up exclusively of small parts. $\bullet$ Outside the central column, only large pieces can be placed. Following the pattern, how many pieces will Nair use to build Figure $20$?

Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?

Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.

A circle of radius $4$ is inscribed in a triangle $ABC$. We call $D$ the touchpoint between the circle and side BC. Let $CD =8$, $DB= 10$. What is the length of the sides $AB$ and $AC$?