CZ-SK-MO 1999

We are allowed to put several brackets in the expression $$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$ always in the same places below each other. (a) Find the smallest possible integer value we can obtain in that way. (b) Find all possible integer values that can be obtained.

Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.

In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$ . Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$ .

Note: Median in a tetrahedron connects a vertex and the centroid of the opposite side.

Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$ , where $t_a,t_b$ are the medians corresponding to $a,b$ , respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$ . Finally, find all possible ratios $a : b$ in such triangles.

In a certain language there are only two letters, $A$ and $B$ . We know that (i) There are no words of length $1$ , and the only words of length $2$ are $AB$ and $BB$ . (ii) A segment of length $n > 2$ is a word if and only if it can be obtained from a word of length less than $n$ by replacing each letter $B$ by some (not necessarily the same) word. Prove that the number of words of length $n$ is equal to $\frac{2^n +2\cdot (-1)^n}{3}$

Given an acute angle $APX$ in the plane, construct a square $ABCD$ such that $P$ lies on the side $BC$ and ray $PX$ meets $CD$ in a point $Q$ such that $AP$ bisects the angle $BAQ$ .

Find all pairs of real numbers $a,b$ for which the system of equations $$\begin{cases} \dfrac{x+y}{x^2 +y^2} = a \dfrac{x^3 +y^3}{x^2 +y^2} = b \end{cases}$$ has a real solution.