During the factoring class, Esmeralda observed that $1$, $3$ and $5$ can be written as the difference of two perfect squares, as can be seen: $1 = 1^2 - 0^2$ $3 = 2^2 - 1^2$ $5 = 3^2 - 2^2$ a) Show that all numbers written in the form $2 * m + 1$ can be written as a difference of two perfect squares. b) Show how to calculate the value of the expression $E = 1 + 3 + 5 + ... + (2m + 1)$. c) Esmeralda, happy with what she discovered, decided to look for other ways to write $2019$ as the difference of two perfect squares of positive integers. Determine how many ways it can do what you want.

Let $ABC$ be a right triangle with hypotenuse $BC$ and center $I$. Let bisectors of the angles $\angle B$ and $\angle C$ intersect the sides $AC$ and $AB$ in$ D$ and $E$, respectively. Let $P$ and $Q$ be the feet of the perpendiculars of the points $D$ and $E$ on the side $BC$. Prove that $I$ is the circumcenter of $APQ$.

We say that a positive integer N is nice if it satisfies the following conditions: $\bullet$ All of its digits are $1$ or $2$ $\bullet$ All numbers formed by $3$ consecutive digits of $N$ are distinct. For example, $121222$ is nice, because the $4$ numbers formed by $3$ consecutive digits of $121222$, which are $121,212,122$ and $222$, are distinct. However, $12121$ is not nice. What is the largest quantity possible number of numbers that a nice number can have? What is the greatest nice number there is?

A positive integer $n$ is called cute when there is a positive integer $m$ such that $m!$ ends in exactly $n$ zeros. a) Determine if $2019$ is cute. b) How many positive integers less than $2019$ are cute?

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $X$ and $K$ points over $AC$ and $AB$, respectively, such that $KX = CX$. Bisector of $\angle AKX$ intersects line $BC$ at $Z$. Show that $XZ$ passes through the midpoint of $BK$.