SE-SKOLORNAS 1983

The positive integers are grouped as follows: $1, 2+3, 4+5+6, 7+8+9+10,\dots$ . Find the value of the $n$ -th sum.

Show that $$\cos x^2 + \cos y^2 - \cos xy < 3$$ for reals $x$ , $y$ .

The systems of equations $$\left\{ \begin{array}{l} 2x_1 - x_2 = 1 -x_1 + 2x_2 - x_3 = 1 -x_2 + 2x_3 - x_4 = 1 -x_3 + 3x_4 - x_5 =1 \cdots\cdots\cdots\cdots -x_{n-2} + 2x_{n-1} - x_n = 1 -x_{n-1} + 2x_n = 1 \end{array} \right.$$ has a solution in positive integers $x_i$ . Show that $n$ must be even.

$C$ , $C'$ are concentric circles with radii $R$ , $R'$ . A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$ . Find its sides if its area is as large as possible.

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$ . What is the smallest possible radius?

Show that the only real solution to $$\left\{ \begin{array}{l} x(x+y)^2 = 9 x(y^3 - x^3) = 7 \end{array} \right.$$ is $x = 1$ , $y = 2$ .