▪ Paraguay Mathematical Olympiad 2012

1. Define a list of number with the following properties:

- The first number of the list is a one-digit natural number.

- Each number (since the second) is obtained by adding to the number before in the list.

- The number is in that list.

Find the first number of the list. 

2. The traveler ant is walking over several chess boards. He only walks vertically and horizontally through the squares of the boards and does not pass two or more times over the same square of a board.

a) In a x board, from which squares can he begin his travel so that he can pass through all the squares of the board?

b) In a x board, from which squares can he begin his travel so that he can pass through all the squares of the board?

c) In a x board, from which squares can he begin his travel so that he can pass through all the squares of the board?

3. Let be a triangle (right in ) inscribed in a semi-circumference of diameter . Determine the distance of the vertice to the side if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle. 

4. Find all four-digit numbers such that they are multiples of and that .

( is a four-digit number; is a two digit-number as is). 

5. Let be an equilateral triangle. Let be a random point in and the point where and the circunscribed circle of intersect.

Prove that

  .

Πηγή: artofproblemsolving