CZ-SK-MO 2002

Solve the system $$(4x)_5+7y=14 (2y)_5 -(3x)_7=74$$ in the domain of integers, where $(n)_k$ stands for the multiple of the number $k$ closest to the number $n$ .

Consider an arbitrary equilateral triangle $KLM$ , whose vertices $K, L$ and $M$ lie on the sides $AB, BC$ and $CD$ , respectively, of a given square $ABCD$ . Find the locus of the midpoints of the sides $KL$ of all such triangles $KLM$ .

Show that a given natural number $A$ is the square of a natural number if and only if for any natural number $n$ , at least one of the differences $$(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A$$ is divisible by $n$ .

Find all pairs of real numbers $a, b$ for which the equation in the domain of the real numbers $$\frac{ax^2-24x+b}{x^2-1}=x$$ has two solutions and the sum of them equals $12$ .

A triangle $KLM$ is given in the plane together with a point $A$ lying on the half-line opposite to $KL$ . Construct a rectangle $ABCD$ whose vertices $B, C$ and $D$ lie on the lines $KM, KL$ and $LM$ , respectively. (We allow the rectangle to be a square.)

Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equality $$f(xf(y))=f(xy)+x$$