BG-MO 1998

Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions.

On the sides of a non-obtuse triangle $ABC$ a square, a regular $n$ -gon and a regular $m$ -gon ( $m$ , $n > 5$ ) are constructed externally, so that their centers are vertices of a regular triangle. Prove that $m = n = 6$ and find the angles of $\triangle ABC$ .

Let $a_1,a_2,\cdots ,a_n$ be real numbers, not all zero. Prove that the equation: $$\sqrt{1+a_1x}+\sqrt{1+a_2x}+\cdots +\sqrt{1+a_nx}=n$$ has at most one real nonzero root.

The sides and diagonals of a regular $n$ -gon $R$ are colored in $k$ colors so that: (i) For each color $a$ and any two vertices $A$ , $B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$ . (ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors. Prove that $k\leq 2$ .