A word is a sequence of capital letters of our alphabet (that is, there are 26 possible letters). A word is called palindrome if has at least two letters and is spelled the same forward and backward. For example, the words "ARARA" e "NOON" are palindromes, but the words "ESMERALDA" and "A" are not palindromes. We say that a word $x$ contains a word $y$ if there are consecutive letters of $x$ that together form the word $y$. For example, the word "ARARA" contains the word "RARA" and also the word "ARARA", but doesn't contain the word "ARRA". Compute the number of words of 14-letter that contain some palindrome.

Show that there are no triples of positive integers $(x,y,z)$ satisfying the equation \[x^2= 5^y+3^z\]

In a triangle scalene $ABC$, let $I$ be its incenter and $D$ the intersection of $AI$ and $BC$. Let $M$ and $N$ points where the incircle touches $AB$ and $AC$, respectively. Let $F$ be the second intersection of the circumcircle $(AMN)$ with the circumcircle $(ABC)$. Let $T$ the intersection of $AF$ and $BC$. Let $J$ be the intersection of $TI$ with the line parallel of $FI$ that passes through $D$. Prove that the line $AJ$ is perpendicular to $BC$.

Find all the positive integers $a,b,c$ such that $3ab= 2c^2$ and $a^3+b^3+c^3$ is the double of a prime number.

The nonzero real numbers $a,b,c$ are such that: $a^2-bc= b^2-ac= c^2-ab= a^3+b^3+c^3$. Compute the possible values of $a+b+c$.

Let $C$ be the set of points $(x,y)$ with integer coordinates in the plane where $1\leq x\leq 900$ and $1\leq y\leq 1000$. A polygon $P$ with vertices in $C$ is called emerald if $P$ has exactly zero or two vertices in each row and each column and all the internal angles of $P$ are $90^\circ$ or $270^\circ$. Find the greatest value of $k$ such that we can color $k$ points in $C$ such that any subset of these $k$ points is not the set of vertices of an emerald polygon. On the left, an example of an emerald polygon; on the right, an example of a non-emerald polygon.

Find all integers $a$ such that there are infinitely many positive integers $n$ such that $n$ divides $\phi(n)!+a$.