IT-CESENATICO 1997

An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?

Let a real function $f$ defined on the real numbers satisfy the following conditions: (i) $f(10+x) = f(10- x)$ (ii) $f(20+x) = - f(20- x)$ for all $x$ . Prove that f is odd and periodic.

The positive quadrant of a coordinate plane is divided into unit squares by lattice lines. Is it possible to color the squares in black and white so that: (i) In every square of side $n$ ( $n \in N$ ) with a vertex at the origin and sides are parallel to the axes, there are more black than white squares; (ii) Every diagonal parallel to the line $y = x$ intersects only finitely many black squares?

Let $ABCD$ be a tetrahedron. Let $a$ be the length of $AB$ and let $S$ be the area of the projection of the tetrahedron onto a plane perpendicular to $AB$ . Determine the volume of the tetrahedron in terms of $a$ and $S$ .

Let $X$ be the set of natural numbers whose all digits in the decimal representation are different. For $n \in N$ , denote by $A_n$ the set of numbers whose digits are a permutation of the digits of $n$ , and $d_n$ be the greatest common divisor of the numbers in $A_n$ . (For example, $A_{1120} =\{112,121,...,2101,2110\}$ , so $d_{1120} = 1$ .) Find the maximum possible value of $d_n$ .

A tourist wants to visit each of the ten cities shown on the picture. The continuous segments on the picture denote railway lines, whereas the dashed segments denote air lines. A railway line costs $150000$ lires, and an air line costs $250000$ lires. What is the minimum possible price of a desired route?

$A$