A natural number is a factorion if it is the sum of the factorials of each of its decimal digits. For example, $145$ is a factorion because $145 = 1! + 4! + 5!$. Find every 3-digit number which is a factorion.

Let $ABC$ be an equilateral triangle with side 3. A circle $C_1$ is tangent to $AB$ and $AC$. A circle $C_2$, with a radius smaller than the radius of $C_1$, is tangent to $AB$ and $AC$ as well as externally tangent to $C_1$. Successively, for $n$ positive integer, the circle $C_{n+1}$, with a radius smaller than the radius of $C_n$, is tangent to $AB$ and $AC$ and is externally tangent to $C_n$. Determine the possible values for the radius of $C_1$ such that 4 circles from this sequence, but not 5, are contained on the interior of the triangle $ABC$.

Let $n$ be a positive integer. A function $f : \{1, 2, \dots, 2n\} \to \{1, 2, 3, 4, 5\}$ is good if $f(j+2)$ and $f(j)$ have the same parity for every $j = 1, 2, \dots, 2n-2$. Prove that the number of good functions is a perfect square.

Let $ABC$ be an acute triangle inscribed on the circumference $\Gamma$. Let $D$ and $E$ be points on $\Gamma$ such that $AD$ is perpendicular to $BC$ and $AE$ is diameter. Let $F$ be the intersection between $AE$ and $BC$. Prove that, if $\angle DAC = 2 \angle DAB$, then $DE = CF$.

Let $n$ be an positive integer and $\sigma = (a_1, \dots, a_n)$ a permutation of $\{1, \dots, n\}$. The cadence number of $\sigma$ is the number of maximal decrescent blocks. For example, if $n = 6$ and $\sigma = (4, 2, 1, 5, 6, 3)$, then the cadence number of $\sigma$ is $3$, because $\sigma$ has $3$ maximal decrescent blocks: $(4, 2, 1)$, $(5)$ and $(6, 3)$. Note that $(4, 2)$ and $(2, 1)$ are decrescent, but not maximal, because they are already contained in $(4, 2, 1)$. Compute the sum of the cadence number of every permutation of $\{1, \dots, n\}$.

Two perfect squares are friends if one is obtained from the other adding the digit $1$ at the left. For instance, $1225 = 35^2$ and $225 = 15^2$ are friends. Prove that there are infinite pairs of odd perfect squares that are friends.

Let $ABC$ be a triangle and $k < 1$ a positive real number. Let $A_1$, $B_1$, $C_1$ be points on the sides $BC$, $AC$, $AB$ such that $$\frac{A_1B}{BC} = \frac{B_1C}{AC} = \frac{C_1A}{AB} = k.$$ (a) Compute, in terms of $k$, the ratio between the areas of the triangles $A_1B_1C_1$ and $ABC$. (b) Generally, for each $n \ge 1$, the triangle $A_{n+1}B_{n+1}C_{n+1}$ is built such that $A_{n+1}$, $B_{n+1}$, $C_{n+1}$ are points on the sides $B_nC_n$, $A_nC_n$ e $A_nB_n$ satisfying $$\frac{A_{n+1}B_n}{B_nC_n} = \frac{B_{n+1}C_n}{A_nC_n} = \frac{C_{n+1}A_n}{A_nB_n} = k.$$Compute the values of $k$ such that the sum of the areas of every triangle $A_nB_nC_n$, for $n = 1, 2, 3, \dots$ is equal to $\dfrac{1}{3}$ of the area of $ABC$.

Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for evert integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times $111$ appears on this sequence?

Let $n$ and $k$ be positive integers. A function $f : \{1, 2, 3, 4, \dots , kn - 1, kn\} \to \{1, \cdots , 5\}$ is good if $f(j + k) - f(j)$ is multiple of $k$ for every $j = 1, 2. \cdots , kn - k$. (a) Prove that, if $k = 2$, then the number of good functions is a perfect square for every positive integer $n$. (b) Prove that, if $k = 3$, then the number of good functions is a perfect cube for every positive integer $n$.

Find every real values that $a$ can assume such that $$\begin{cases} x^3 + y^2 + z^2 = a\\ x^2 + y^3 + z^2 = a\\ x^2 + y^2 + z^3 = a \end{cases}$$has a solution with $x, y, z$ distinct real numbers.

Let $\Theta_1$ and $\Theta_2$ be circumferences with centers $O_1$ and $O_2$, exteriorly tangents. Let $A$ and $B$ be points in $\Theta_1$ and $\Theta_2$, respectively, such that $AB$ is common external tangent to $\Theta_1$ and $\Theta_2$. Let $C$ and $D$ be points on the semiplane determined by $AB$ that does not contain $O_1$ and $O_2$ such that $ABCD$ is a square. If $O$ is the center of this square, compute the possible values for the angle $\angle O_1OO_2$.