AUSTRIAN-POLISH 1993

Solve in positive integers $x,y$ the equation $2^x - 3^y = 7$ .

Consider all tetrahedra $ABCD$ in which the sum of the areas of the faces $ABD, ACD, BCD$ does not exceed $1$ . Among such tetrahedra, find those with the maximum volume.

Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ( $r > 1, k_i > 0$ ). Given $m > 1$ , we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge 0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$ , determine $g(m)$ .

The Fibonacci numbers are defined by $F_0 = 1, F_1 = 1, F_{n+2} = F_{n+1} + F_n$ . The positive integers $A, B$ are such that $A^{19}$ divides $B^{93}$ and $B^{19}$ divides $A^{93}$ . Show that if $h < k$ are consecutive Fibonacci numbers then $(AB)^h$ divides $(A^4 + B^8)^k$

Solve in real numbers the system $$\begin{cases} x^3 + y = 3x + 4 2y^3 + z = 6y + 6 3z^3 + x = 9z + 8\end{cases}$$

If $a,b \ge 0$ are real numbers, prove the inequality $$\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2\leq\frac{a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b}{4}\leq\frac{a+\sqrt{ab}+b}{3} \leq \sqrt{\left(\frac{a^{2/3}+b^{2/3}}{2}\right)^{3}}$$ For each of the inequalities, find the cases of equality.

The sequence $(a_n)$ is defined by $a_0 = 0$ and $a_{n+1} = [\sqrt[3]{a_n +n}]^3$ for $n \ge 0$ . (a) Find $a_n$ in terms of $n$ . (b) Find all $n$ for which $a_n = n$ .

Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions $Q(0)= 0$ , $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$ .

Point $P$ is taken on the extension of side $AB$ of an equilateral triangle $ABC$ so that $A$ is between $B$ and $P$ . Denote by $a$ the side length of triangle $ABC$ , by $r_1$ the inradius of triangle $PAC$ , and by $r_2$ the exradius of triangle $PBC$ opposite $P$ . Find the sum $r_1+r_2$ as a function in $a$ .