1993 Nordic

Part of Nordic

Subcontests

(4)

1

F(x/3) = F(x)/2, F(1-x)=1-F(x), F( 173/1993) ?, F(1/13)?

F

be an increasing real function defined for all

x, 0 \le x \le 1

, satisfying the conditions (i)

F (\frac{x}{3}) = \frac{F(x)}{2}

F(1- x) = 1 - F(x)

F(\frac{173}{1993})

F(\frac{1}{13})

1

sum of digits of decimal representation of n, T(1993N),T(kN)

T(n)

the sum of the digits of the decimal representation of a positive integer

n

. a) Find an integer

N

T(k \cdot N)

is even for all

k, 1 \le k \le 1992,

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

1

system 3x3 with integers and numbers of their digits

Find all solutions of the system of equations

\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}

x, y

z

are positive integers, and

s(x), s(y)

s(z)

are the numbers of digits in the decimal representations of

x, y

z

, respectively.

1

hexagon with sides 1,1,2,2,3,3 find it's circumradius

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

r

satisfies the equation

2r^3 - 7r - 3 = 0