▪ Brazil Olympic Revenge 2012

1. Let and real numbers. Let a continuous function. We say that f is "smp" if satisfying and for each :

or 

Prove that if is continuous such that for each there are only finitely many satisfying , then is "smp".

2. We define

  , 

where the indices are taken modulo .

Besides this, if is a vector, we define , if , or , otherwise.

Prove that, if , for some natural number , then .

3. Let be a finite graph. Prove that one can partition into two graphs such that if we erase all edges conecting a vertex from to a vertex from , each vertex of the new graph has even degree. 

4. Say that two sets of positive integers are if the sum of the th powers of elements of equals the sum of the th powers of elements of , for each . Given , prove that there are infinitely many numbers such that can be divided into subsets, all of which are -equivalent to each other. 

5. Let positive real numbers. Prove that:

6. Let be an scalene triangle and and its incenter, ortocenter respectively.

The incircle touchs , and at an . and intersects at while and intersets at .

Shows that cannot be paralel to .