NMC 1993

Let $F$ be an increasing real function defined for all $x, 0 \leq x \leq 1$, satisfying the conditions

$$\begin{aligned} & F\left(\frac{x}{3}\right)=\frac{F(x)}{2} \\ & F(1-x)=1-F(x) \end{aligned}$$

Determine $F\left(\frac{173}{1993}\right)$ and $F\left(\frac{1}{13}\right)$.

A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length 1, two have length 2 and two have length 3. Show that $r$ satisfies the equation

$$2 r^{3}-7 r-3=0$$

Figure 5.

Find all solutions of the system of equations

$$\left\{\begin{aligned} s(x)+s(y) & =x \\ x+y+s(z) & =z \\ s(x)+s(y)+s(z) & =y-4 \end{aligned}\right.$$

where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.

Denote by $T(n)$ the sum of the digits of the decimal representation of a positive integer $n$.

a) Find an integer $N$, for which $T(k \cdot N)$ is even for all $k, 1 \leq k \leq 1992$, but $T(1993 \cdot N)$ is odd. b) Show that no positive integer $N$ exists such that $T(k \cdot N)$ is even for all positive integers $k$.