SE-SKOLORNAS 2008

A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$ . Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$ .

Determine the smallest integer $n \ge 3$ with the property that you can choose two of the numbers $1,2,\dots, n$ in such a way that their product is equal to the sum of the other $n - 2$ languages. What are the two numbers?

The function $f(x)$ has the property that $\frac{f(x)}{x}$ is increasing for $x>0$ . Show that $$f(x)+f(y) \leq f(x+y) , \text{for all } x,y>0$$

A convex $n$ -side polygon has angles $v_1,v_2,\dots,v_n$ (in degrees), where all $v_k$ ( $k = 1,2,\dots,n$ ) are positive integers divisible by $36$ . (a) Determine the largest $n$ for which this is possible. (b) Show that if $n>5$ , two of the sides of the $n$ -polygon must be parallel.

Anna and Orjan play the following game: they start with a positive integer $n>1$ , Anna writes it as the sum of two other positive integers, $n = n_1+n_2$ . Orjan deletes one of them, $n_1$ or $n_2$ . If the remaining number is larger than $1$ , the process is repeated, i.e. Anna writes it as the sum of two positive integers, $n_3+n_4$ , Orjan deletes one of them etc. The game ends when the last number is $1$ . Orjan is the winner if there are two equal numbers among the numbers he has deleted, otherwise Anna wins. Who is winning the game if n = 2008 and they both play optimally?

A sum decomposition of the number 100 is given by a positive integer $n$ and $n$ positive integers $x_1<x_2<\cdots <x_n$ such that $x_1 + x_2 + \cdots + x_n = 100$ . Determine the largest possible value of the product $x_1x_2\cdots x_n$ , and $n$ , as $x_1, x_2,\dots, x_n$ vary among all sum decompositions of the number $100$ .