PE-MO 2012

For each positive integer $n$ whose canonical decomposition is $n = p_1^{a_1} \cdot p_2^{a_2} \cdot\cdot\cdot p_k^{a_k}$ , we define $t(n) = (p_1 + 1) \cdot (p_2 + 1) \cdot\cdot\cdot (p_k + 1)$ . For example, $t(20) = t(2^2\cdot 5^1) = (2 + 1) (5 + 1) = 18$ , $t(30) = t(2^1\cdot 3^1\cdot 5^1) = (2 + 1) (3 + 1) (5 + 1) = 72$ and $t(125) = t(5^3) = (5 + 1) = 6$ .

We say that a positive integer $n$ is specialif $t(n)$ is a divisor of $n$ . How many positive divisors of the number $54610$ are special?

Let $x, y$ be nonzero real numbers that satisfy the following equation: $$x^3 + y^3 + 3x^2y^2 = x^3y^3.$$ Determine all the values that the expression $\frac{1}{x}+ \frac{1}{y}$ can take.

A domino is a $1\times2$ or $2\times 1$ rectangle. Diego wants to completely cover a $6\times 6$ board using $18$ dominoes. Determine the smallest positive integer $k$ for which Diego can place $k$ dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remaining dominoes.