SE-SKOLORNAS 1993

An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$ .

A railway line is divided into ten sections by the stations $A,B,C,D,E,F, G,H,I,J,K$ . The length of each section is an integer number of kilometers and the distacne between $A$ and $K$ is $56$ km. A trip along two successive sections never exceeds $12$ km, but a trip along three successive sections is at least $17$ km. What is the distance between $B$ and $G$ ?

Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.

To each pair of nonzero real numbers $a$ and $b$ a real number $a*b$ is assigned so that $a*(b*c) = (a*b)c$ and $a*a = 1$ for all $a,b,c$ . Solve the equation $x*36 = 216$ .

A triangle with sides $a,b,c$ and perimeter $2p$ is given. Is possible, a new triangle with sides $p-a$ , $p-b$ , $p-c$ is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely?

For real numbers $a$ and $b$ define $f(x) = \frac{1}{ax+b}$ . For which $a$ and $b$ are there three distinct real numbers $x_1,x_2,x_3$ such that $f(x_1) = x_2$ , $f(x_2) = x_3$ and $f(x_3) = x_1$ ?