CL-MO 2013

Find the sum of all $5$ -digit positive integers that they have only the digits $1, 2$ , and $5$ , none repeated more than three consecutive times.

Hannibal and Clarice are still at a barbecue and there are three anticuchos left, each of which it has $10$ pieces. Of the $30$ total pieces, there are $29$ chicken and one meat, the which is at the bottom of one of the anticuchos. To decide who to stay with the piece of meat, they decide to play the following game: they alternately take out a piece of one of the anticuchos (they can take only the outer pieces) and whoever wins the game manages to remove the piece of meat. Clarice decides if she starts or Hannibal starts. What should she decide?

Given a finite sequence of real numbers $a_1,a_2,...,a_n$ such that $$a_1 + a_2 + ... + a_n > 0.$$ Prove that there is at least one index $i$ such that $$a_i > 0, a_i + a_{i+1} > 0, ..., a_i + a_{i+1} + ...+ a_n > 0.$$

Consider a function f defined on the positive integers that meets the following conditions: $$f(1) = 1 , \, f(2n) = 2f(n) , \, nf(2n + 1) = (2n + 1)(f(n) + n)$$ for all $n \ge 1$ . a) Prove that $f(n)$ is an integer for all $n$ . b) Find all positive integers $m$ less than $2013$ that satisfy the equation $f(m) = 2m$ .