CZ-SK-MO 1998

Solve the equation $x\cdot [x\cdot [x \cdot [x]]] = 88$ in the set of real numbers.

Given any set of $14$ (different) natural numbers, prove that for some $k$ ( $1 \le k \le 7$ ) there exist two disjoint $k$ -element subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$ such that $A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}$ and $B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}$ differ by less than $0.001$ , i.e. $|A-B| < 0.001$

A sphere is inscribed in a tetrahedron $ABCD$ . The tangent planes to the sphere parallel to the faces of the tetrahedron cut off four smaller tetrahedra. Prove that sum of all the $24$ edges of the smaller tetrahedra equals twice the sum of edges of the tetrahedron $ABCD$ .

For each date of year $1998$ , we calculate day $^{month}$ -year and determine the greatest power of $3$ that divides it. For example, for April $21$ we get $21^4 - 1998 =192483 = 3^3 \cdot 7129$ , which is divisible by $3^3$ and not by $3^4$ . Find all dates for which this power of $3$ is the greatest.

A circle $k$ and a point $A$ outside it are given in the plane. Prove that all trapezoids, whose non-parallel sides meet at $A$ , have the same intersection of diagonals.

Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations $$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.