▪ Austria Federal Competition For Advanced Students 2012, Part 2

Ημέρα 1η

1. Determine the maximum value of , such that the inequality

holds for every with

  .

When does equality occur? 

2. Solve over :

3. We call an isosceles trapezoid interesting, if it is inscribed in the unit square in such a way, that on every side of the square lies exactly one vertex of the trapezoid and that the lines connecting the midpoints of two adjacent sides of the trapezoid are parallel to the sides of the square.

Find all interesting isosceles trapezoids and their areas. 

Ημέρα 2η

1.. Given a sequence of real numbers, we define as the arithmetic mean of the numbers to for .

If there is a real number , such that

for every triple of distinct positive integers, prove that the sequence is an arithmetic progression. 

2. We define as the set of natural numbers with the following property:

There exists an integer exponent with , such that . Find . 

3. Given an equilateral triangle with sidelength 2, we consider all equilateral triangles with sidelength 1 such that:

lies on the side ,

lies on the side , and

lies in the inside or on the perimeter of .

Find the locus of the centroids of all such triangles .

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